Path Integrals in Polymer Physics

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H. Kleinert, PATH INTEGRALS January 6, 2016 (/home/kleinert/kleinert/books/pathis/pthic15.tex)Thou com’st in such a questionable shape, That I will speak to thee. William Shakespeare (1564–1616), Hamlet

15 Path Integrals in Polymer Physics

The use of path integrals is not confined to the quantum-mechanical description point particles in spacetime. An important field of applications lies in polymer physics where they are an ideal tool for studying the statistical fluctuations of line- like physical objects.

15.1 Polymers and Ideal Random Chains

A polymer is a long chain of many identical molecules connected with each other at joints which allow for spatial rotations. A large class of polymers behaves approximately like an idealized random chain. This is defined as a chain of N links of a fixed length a, whose rotational angles occur all with equal probability (see Fig. 15.1). The probability distribution of the end-to-end distance vector xb xa of such an object is given by − N 1 N P (x x )= d3∆x δ( ∆x a) δ(3)(x x ∆x ). (15.1) N b − a n 4πa2 | n| − b − a − n nY=1 Z nX=1 The last δ-function makes sure that the vectors ∆xn of the chain elements add up correctly to the distance vector x x . The δ-functions under the product enforce b − a the fixed length of the chain elements. The length a is also called the bond length of the random chain. The angular probabilities of the links are spherically symmetric. The factors 1/4πa2 ensure the proper normalization of the individual one-link probabilities 1 P (∆x)= δ( ∆x a) (15.2) 1 4πa2 | | − in the integral

3 d xb P1(xb xa)=1. (15.3) Z − The same normalization holds for each N:

3 d xb PN (xb xa)=1. (15.4) Z − 1031 1032 15 Path Integrals in Polymer Physics

Figure 15.1 Random chain consisting of N links ∆xn of length a connecting xa = x0 and xb = xN .

If the second δ-function in (15.1) is Fourier-decomposed as

N 3 d k N (3) ik(xb−xa)−ik ∆xn δ xb xa ∆xn = e n=1 , (15.5) − − ! (2π)3 nX=1 Z P we see that P (x x ) has the Fourier representation N b − a d3k x x ik(xb−xa) ˜ k PN ( b a)= 3 e PN ( ), (15.6) − Z (2π) with N 1 P˜ (k)= d3∆x δ( ∆x a)e−ik∆xn . (15.7) N n 4πa2 | n| − nY=1 Z

Thus, the Fourier transform P˜N (k) factorizes into a product of N Fourier- transformed one-link probabilities:

N P˜N (k)= P˜1(k) . (15.8) h i These are easily calculated: 1 sin ka ˜ k 3 x x −ik∆x P1( )= d ∆ 2 δ( ∆ a)e = . (15.9) Z 4πa | | − ka The desired end-to-end probability distribution is then given by the integral d3k R ˜ k N ikR PN ( ) = 3 [P1( )] e Z (2π) 1 ∞ sin ka N = 2 dkk sin kR , (15.10) 2π R Z0 " ka #

H. Kleinert, PATH INTEGRALS 15.2 Moments of End-to-End Distribution 1033 where we have introduced the end-to-end distance vector

R x x . (15.11) ≡ b − a The generalization of the one-link distribution to D dimensions is 1 x x P1(∆ )= D−1 δ( ∆ a), (15.12) SDa | | − with SD being the surface of a unit sphere in D dimensions [see Eq. (1.558)]. To calculate 1 ˜ k D x x −ik∆x P1( )= d ∆ D−1 δ( ∆ a)e , (15.13) Z SDa | | − we insert for e−ik∆x = e−ik|∆x| cos ∆ϑ the expansion (8.130) with (8.101), and use Y0,0(xˆ)=1/√SD to perform the integral as in (8.250). Using the relation between 1 modiﬁed and ordinary Bessel functions Jν(z)

−iπ/2 −iπ/2 Iν (e z)= e Jν(z), (15.14) this gives Γ(D/2) P˜ (k)= J (ka), (15.15) 1 (ka/2)D/2−1 D/2−1 where Jµ(z) is the Bessel function.

15.2 Moments of End-to-End Distribution

The end-to-end distribution of a random chain is, of course, invariant under rotations so that the Fourier-transformed probability can only have even Taylor expansion coefficients: ∞ (ka)2l P˜ (k) = [P˜ (k)]N = P . (15.16) N 1 N,2l (2l)! Xl=0 The expansion coefficients provide us with a direct measure for the even moments of the end-to-end distribution. These are defined by

2l D 2l R d R R PN (R). (15.17) h i ≡ Z 2l The relation between R and PN,2l is found by expanding the exponential under the inverse of the Fourierh integrali (15.6): ∞ ( ikR)n P˜ (k)= dDR e−ikRP (R)= dDR − P (R), (15.18) N N n! N Z nX=0 Z 1I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.406.1. 1034 15 Path Integrals in Polymer Physics and observing that the angular average of (kR)n is related to the average of Rn in D dimensions by 0 , n = odd, (kR)n = kn Rn (n 1)!!(D 2)!! (15.19) h i h i  − − , n = even.  (D + n 2)!! − The three-dimensional result 1/(n+1) follows immediately from the angular average 1 n (1/2) −1 d cos θ cos θ being 1/(n + 1) for even n. In D dimensions, it is most easily derivedR by assuming, for a moment, that the vectors R have a Gaussian distribution (0) 2 3/2 −R2D/2Na2 PN (R)=(D/2πNa ) e . Then the expectation values of all products of Ri can be expressed in terms of the pair expectation value 1 R R (0) = δ a2N (15.20) h i ji D ij via Wick’s rule (3.305). The result is 1 R R R (0) = δ anN n/2, (15.21) h i1 i2 ··· in i Dn/2 i1i2i3...in with the contraction tensor δi1i2i3...in of Eqs. (8.64) and (13.89), which has the recursive deﬁnition

δi1i2i3...in = δi1i2 δi3i4...in + δi1i3 δi2i4...in + ...δi1in δi2i3...in−1 . (15.22) A full contraction of the indices gives, for even n, the Gaussian expectation values:

(D + n 2)!! Γ(D/2+ n/2) 2n/2 Rn (0) = − anN n/2 = anN n/2, (15.23) h i (D 2)!!Dn/2 Γ(D/2) Dn/2 − for instance

(0) (D + 2) (0) (D +2)(D + 4) R4 = a4N 2, R6 = a6N 3. (15.24) D D2 D E D E By contracting (15.21) with k k k we ﬁnd i1 i2 ··· in 1 (kR)n (0) =(n 1)!! (ka)nN n/2 = kn (R)n (0) d , (15.25) h i − Dn/2 h i n with (n 1)!!(D 2)!! d = − − . (15.26) n (D + n 2)!! − Relation (15.25) holds for any rotation-invariant size distribution of R, in particular for PN (R), thus proving Eq. (15.17) for random chains. Hence, the expansion coeﬃcients P are related to the moments R2l by N,2l h iN P =( 1)ld R2l , (15.27) N,2l − 2lh i

H. Kleinert, PATH INTEGRALS 15.2 Moments of End-to-End Distribution 1035 and the moment expansion (15.16) becomes

∞ ( 1)l(k)2l P˜ (k)= − d R2l . (15.28) N (2l)! 2l h i Xl=0 Let us calculate the even moments R2l of the polymer distribution P (R) h i N explicitly for D = 3. We expand the logarithm of the Fourier transform P˜N (k) as follows:

∞ 2l l sin ka 2 ( 1) B2l 2l log P˜N (k)= N log P˜1(k)= N log = N − (ka) , (15.29) ka ! (2l)!2l Xl=1 where B are the Bernoulli numbers B =1/6, B = 1/30,... . Then we note that l 2 4 − for a Taylor series of an arbitrary function y(x)

∞ a y(x)= n xn, (15.30) n! nX=1 the exponential function ey(x) has the expansion

∞ b ey(x) = n xn, (15.31) n! nX=1 with the coeﬃcients

b n 1 a mi n = i . (15.32) n! mi! i! {Xmi} iY=0

The sum over the powers mi =0, 1, 2,... obeys the constraint

n n = i m . (15.33) · i Xi=1

Note that the expansion coefficients an of y(x) are the cumulants of the expansion y(x) coefficients bn of e as defined in Section 3.17. For the coefficients an of the expansion (15.29),

2l l N2 ( 1) B2l/2l for n =2l, an = − − (15.34) ( 0 for n =2l +1, we ﬁnd, via the relation (15.27), the moments

l 1 N22i( 1)iB mi R2l = a2l( 1)l(2l + 1)! − 2i , (15.35) h i − mi! " (2i)!2i # {Xmi} iY=1 with the sum over mi =0, 1, 2,... constrained by 1036 15 Path Integrals in Polymer Physics

l l = i m . (15.36) · i Xi=1 For l = 1 and 2 we obtain the moments 5 2 R2 = a2N, R4 = a4N 2 1 . (15.37) h i h i 3 − 5N In the limit of large N, the leading behavior of the moments is the same as in (15.20) and (15.24). The linear growth of R2 with the number of links N is characteristic for a random chain. In the presenceh ofi interactions, there will be a diﬀerent power behavior expressed as a so-called scaling law

R2 a2N 2ν . (15.38) h i ∝ The number ν is called the critical exponent of this scaling law. It is intuitively obvious that ν must be a number between ν =1/2 for a random chain as in (15.37), and ν = 1 for a completely stiﬀ chain. Note that the knowledge of all moments of the end-to-end distribution determines completely the shape of the distribution by an expansion

1 ∞ ( 1)n R n n PL( )= D−1 R − ∂Rδ(R). (15.39) SDR h i n! nX=0 This can easily be veriﬁed by calculating the integrals (15.17) using the integrals formula

n n n dz z ∂z δ(z)=( 1) n! . (15.40) Z − 15.3 Exact End-to-End Distribution in Three Dimensions

Consider the Fourier representation (15.10), rewritten as

i ∞ sin η N R −iηR/a PN ( )= 2 2 dηη e , (15.41) 4π a R Z−∞ η ! with the dimensionless integration variable η ka. By expanding ≡ 1 N N sinN η = ( 1)n exp [i(N 2n)η] , (15.42) (2i)N − n ! − nX=0 we ﬁnd the ﬁnite series

N 1 n N PN (R)= ( 1) IN (N 2n R/a), (15.43) 2N+2iN−1π2a2R − n ! − − nX=0

H. Kleinert, PATH INTEGRALS 15.3 Exact End-to-End Distribution in Three Dimensions 1037

where IN (x) are the integrals ∞ eiηx IN (x) dη N−1 . (15.44) ≡ Z−∞ η For N 2, these integrals are all singular. The singularity can be avoided by noting ≥ that the initial integral (15.41) is perfectly regular at η = 0. We therefore replace the expression (sin η/η)N in the integrand by [sin(η iǫ)/(η iǫ)]N . This regularizes each term in the expansion (15.43) and leads to well-deﬁned− − integrals:

∞ eix(η−iǫ) IN (x)= dη . (15.45) −∞ (η iǫ)N−1 Z − For x< 0, the contour of integration can be closed by a large semicircle in the lower half-plane. Since the lower half-plane contains no singularity, the residue theorem shows that

IN (x)=0, x< 0. (15.46)

For x > 0, on the other hand, an expansion of the exponential function in powers of η−iǫ produces a pole, and the residue theorem yields 2πiN−1 I (x)= xN−2, x> 0. (15.47) N (N 2)! − Hence we arrive at the ﬁnite series

1 n N N−2 PN (R)= ( 1) (N 2n R/a) . (15.48) 2N+1(N 2)!πa2R − n ! − − − 0≤n≤(NX−R/a)/2 The distribution is displayed for various values of N in Fig. 15.2, where we have 2 plotted 2πR √NPN (R) against the rescaled distance variable ρ = R/√Na. With this N-dependent rescaling all curves have the same unit area. Note that they converge rapidly towards a universal zero-order distribution

3 D 2 (0) 3 3R (0) D −DR2/2aL PN (R)= exp PL (R)= e . (15.49) s2πNa2 (−2Na2 ) → s2πaL In the limit of large N, the length L will be used as a subscript rather than the diverging N. The proof of the limit is most easily given in Fourier space. For 2 2 large N at ﬁnite k a N, the Nth power of the quantity P˜1(k) in Eq. (15.15) can be approximated by

2 2 [P˜ (k)]N e−Nk a /2D. (15.50) 1 ∼ Then the Fourier transform (15.10) is performed with the zero-order result (15.49). In Fig. 15.2 we see that this large-N limit is approached uniformly in ρ = R/√Na. The approach to this limit is studied analytically in the following two sections. 1038 15 Path Integrals in Polymer Physics

Figure 15.2 End-to-end distribution PN (R) of random chain with N links. The func- 2 tions 2πR √NPN (R) are plotted against R/√N a which gives all curves the same unit area. Note the fast convergence for growing N. The dashed curve is the continuum distri- (0) bution PL (R) of Eq. (15.49). The circles on the abscissa mark the maximal end-to-end distance.

15.4 Short-Distance Expansion for Long Polymer

At ﬁnite N, we expect corrections which are expandable in the form

∞ (0) 1 2 2 PN (R)= PN (k) 1+ Cn(R /Na ) , (15.51) " N n # nX=1 0 where the functions Cn(x) are power series in x starting with x . Let us derive this expansion. In three dimensions, we start from (15.29) and separate the right-hand side into the leading k2-term and a remainder

∞ 22l( 1)lB C˜(k) exp N − 2l (k2a2)l . (15.52) ≡ " (2l)!2l # Xl=2 Exponentiating both sides of (15.29), the end-to-end probability factorizes as

−Na2k2/6 P˜N (k)= e C˜(k). (15.53)

The function C˜(k) is now expanded in a power series

n 2 2 l C˜(k)=1+ C˜n,lN (a k ) , (15.54) n=1,X2,... l=2n,2n+1,...

H. Kleinert, PATH INTEGRALS 15.4 Short-Distance Expansion for Long Polymer 1039 with the lowest coeﬃcients 1 1 C˜ = , C˜ = , 1,2 −180 1,3 −2835 1 1 C˜ = , C˜ = ,.... (15.55) 1,4 −37800 2,4 64800 For any dimension D, we factorize

−Na2k2/2D PÑ (k)= e C˜(k) (15.56) and find C˜(k) by expanding (15.15) in powers of k and proceeding as before. This gives the coefficients

2 C˜1,2 = 1/4D (D + 2), − 3 C˜1,3 = 1/3D (D + 2)(D + 4), − 4 2 C˜1,4 = (5D + 12)/8D (D + 2) (D + 4)(D + 6), − 4 2 C˜2,4 = 1/32D (D + 2) . (15.57) We now Fourier-transform (15.56). The leading term in C˜(k) yields the zero-order distribution (15.49) in D dimensions,

D D 2 2 P (0)(R)= e−DR /2Na , (15.58) N s2πNa2 or, written in terms of the reduced distance variable ρ = R/Na,

D D 2 P (0)(R)= e−DNρ /2. (15.59) N s2πNa2 To account for the corrections in C˜(k) we take the expansion (15.54), emphasize the dependence on k2a2 by writing C˜(k)= C¯(k2a2), and observe that in the Fourier transform 3 3 d k d k 2 2 R ikR k ikR −Nk a /2D ¯ 2 2 PN ( )= 3 e PN ( )= 3 e e C(k a ), (15.60) Z (2π) Z (2π) the series can be pulled out of the integral by replacing each power (k2a2)p by ( 2D∂ )p. The result has the form − N 3 d k 2 2 R ¯ ikR −Nk a /2D ¯ (0) R PN ( )= C( 2D∂N ) 3 e e = C( 2D∂N )PN ( ). (15.61) − Z (2π) − Going back to coordinate space, we obtain an expansion

D/2 D 2 R −DNρ /2 PN ( )= 2 e C(R). (15.62) 2πNa 1040 15 Path Integrals in Polymer Physics

For D = 3, the function C(R) is given by the series

∞ −n 2 l C(R)=1+ Cn,lN (Nρ ) , (15.63) n=1 l=0X,...,2n with the coefficients 3 3 9 29 69 981 1341 81 C1,l = , , , C2,l = , , , , . (15.64) −4 2 −20 160 −40 400 −1400 800 For any D, we find the coefficients D D D2 C1,l = , , , − 4 2 −4(D + 2)! (3D2 2D + 8)D (D2 +2D + 8)D (3D2 + 14D + 40)D2 C2,l = − , , , 96(D + 2) − 8(D + 2) 16(D + 2)2 (3D2 + 22D + 56)D3 D4 , . (15.65) − 24(D + 2)2(D + 4) 32(D + 2)2 !

15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution

Another study of the approach to the limiting distribution (15.49) proceeds via the saddle point approximation. For this, the integral in (15.41) is rewritten as

∞ dηηe−Nf(η), (15.66) Z−∞ with R sin η f(η)= i η log . (15.67) Na − η ! The extremum of f lies at η =η ¯, whereη ¯ solves the equation 1 R coth(iη¯) = . (15.68) − iη¯ Na The function on the left-hand side is known as the Langevin function: 1 L(x) coth x . (15.69) ≡ − x The extremum lies at the imaginary positionη ¯ ix¯ withx ¯ being determined by the equation ≡ − R L(¯x)= . (15.70) Na

H. Kleinert, PATH INTEGRALS 15.6 Path Integral for Continuous Gaussian Distribution 1041

The extremum is a minimum of f(η) since 1 1 f ′′(¯η)= L′(¯x)= + > 0. (15.71) −sinh2 x¯ x¯2 By shifting the integration contour vertically into the complex η plane to make it run through the minimum at ix¯, we obtain − 1 ∞ N R −Nf(¯η) ′′ 2 PN ( ) 2 2 e dη ( ix¯ + η) exp f (¯η)η ≈ −4iπ a R Z−∞ − − 2 η¯ 2π = e−Nf(¯η). (15.72) 4π2a2R sNf ′′(¯x)

When expressed in terms of the reduced distance ρ R/Na [0, 1], this reads ≡ ∈ 1 Li(ρ)2 sinh Li(ρ) N PN (R) . (15.73) ≈ (2πNa2)3/2 ρ 1 [Li(ρ)/ sinh Li(ρ)]2 1/2 (Li(ρ) exp[ρLi(ρ)]) { − } Here we have introduced the inverse Langevin function Li(ρ) since it allows us to expressx ¯ as

x¯ = Li(ρ) (15.74)

[inverting Eq. (15.70)]. The result (15.73) is valid in the entire interval ρ [0, 1] ∈ corresponding to R [0, Na]; it ignores corrections of the order 1/N. By expanding the right-hand side∈ in a power series in ρ, we ﬁnd

3 3/2 3R2 3R2 9R4 R PN ( )= 2 exp 2 1+ 2 2 3 4 + ... , (15.75) N 2πNa −2Na ! 2N a − 20N a ! with some normalization constant . At each order of truncation, is determined in such a way that d3R P (R)=N 1. As a check we take the limitNρ2 1/N and N ≪ ﬁnd powers in ρ whichR agree with those in (15.62), for D = 3, with the expansion (15.63) of the correction factor.

15.6 Path Integral for Continuous Gaussian Distribution

The limiting end-to-end distribution (15.49) is equal to the imaginary-time ampli- tude of a free particle in natural units withh ¯ = 1:

2 1 M (xb xa) (xbτb xaτa)= D exp − . (15.76) | 2π(τ τ )/M "− 2 τb τa # b − a − q We merely have to identify

x x R, (15.77) b − a ≡ 1042 15 Path Integrals in Polymer Physics and replace

τ τ Na, (15.78) b − a → M D/a. (15.79) → Thus we can describe a polymer with R2 Na2 by the path integral ≪ L D D ′ 2 D −DR2/2La PL(R)= x exp ds [x (s)] = e . (15.80) s Z D (−2a Z0 ) 2πa The number of time slices is here N [in contrast to (2.66) where it was N +1), and the total length of the polymer is L = Na. Let us calculate the Fourier transformation of the distribution (15.80):

D −i q·R P˜L(q)= d R e PL(R). (15.81) Z After a quadratic completion the integral yields

−Laq2/2D P˜L(q)= e , (15.82) with the power series expansion

∞ q2l La l P˜ (q)= ( 1)l . (15.83) L − l! 2D Xl=0 Comparison with the moments (15.23) shows that we can rewrite this as

∞ q2l Γ(D/2) P˜ (q)= ( 1)l R2l . (15.84) L − l! 22lΓ(D/2+ l) Xl=0 D E This is a completely general relation: the expansion coeﬃcients of the Fourier transform yield directly the moments of a function, up to trivial numerical factors speciﬁed by (15.84). The end-to-end distribution determines rather directly the structure factor of a dilute solution of polymers which is observable in static neutron and light scattering experiments:

1 L L ′ S(q)= ds ds′ eiq·[x(s)−x(s )] . (15.85) L2 0 0 Z Z D E The average over all polymers running from x(0) to x(L) can be written, more explicitly, as

′ eiq·[x(s)−x(s )] = dDx(L) dD(x(s′) x(s)) dDx(0) (15.86) − Z Z ′ Z D E ′ −iq·x(s ) ′ iq·x(s) P ′ (x(L) x(s ))e P ′ (x(s ) x(s))e P (x(s) x(0)). × L−s − s −s − s−0 −

H. Kleinert, PATH INTEGRALS 15.6 Path Integral for Continuous Gaussian Distribution 1043

The integrals over initial and ﬁnal positions give unity due to the normalization (15.4), so that we remain with

iq·[x(s)−x(s′)] D −iq·R e = d R e Ps′−s(R). (15.87) D E Z Since this depends only on L′ s′ s and not on s + s′, we decompose the double integral in over s and s′ in (15.85)≡| into− | 2 L dL′(L L′) and obtain 0 − L R 2 ′ ′ D iq·R(L′) q ′ R S( )= 2 dL (L L ) d R e PL ( ), (15.88) L Z0 − Z or, recalling (15.81),

L 2 ′ ′ q ˜ ′ q S( )= 2 dL (L L )PL ( ). (15.89) L Z0 − Inserting (15.82) we obtain the Debye structure factor of Gaussian random paths:

2 q2aL SGauss(q)= x 1+ e−x , x . (15.90) x2 − ≡ 2D This function starts out like 1 x/3+ x2/12 + ... for small q and falls of like q−2 for q2 2D/aL. The Taylor− coeﬃcients are determined by the moments of the ≫ end-to-end distribution. By inserting (15.84) into (15.89) we obtain:

∞ Γ(D/2) 2 L S(q)= ( 1)lq2l dL′(L L′) R2l . (15.91) − 22ll!Γ(l + D/2) L2 0 − Xl=0 Z D E Although the end-to-end distribution (15.80) agrees with the true polymer distribution (15.1) for R √Na, it is important to realize that the nature of the fluctuations in the two expressions≪ is quite different. In the polymer expression, the length of each link ∆xn is fixed. In the sliced action of the path integral Eq. (15.80),

N M (∆x )2 N = a n , (15.92) A 2 a2 nX=1 on the other hand, each small section ﬂuctuates around zero with a mean square

a a2 (∆x )2 = = . (15.93) h n i0 M D Yet, if the end-to-end distance of the polymer is small compared to the completely stretched configuration, the distributions are practically the same. There exists a qualitative difference only if the polymer is almost completely stretched. While the polymer distribution vanishes for R>Na, the path integral (15.80) gives a nonzero value for arbitrarily large R. Quantitatively, however, the difference is insignificant since it is exponentially small (see Fig. 15.2). 1044 15 Path Integrals in Polymer Physics

15.7 Stiﬀ Polymers

The end-to-end distribution of real polymers found in nature is never the same as that of a random chain. Usually, the joints do not allow for an equal probability of all spherical angles. The forward angles are often preferred and the polymer is stiff at shorter distances. Fortunately, if averaged over many links, the effects of the stiffness becomes less and less relevant. For a very long random chain with a finite stiffness one finds the same linear dependence of the square end-to-end distance on the length L = Na as for ideal random chains which has, according to Eq. (15.23), the Gaussian expectations: (D +2l 2)!! R2 = aL, R2l = − (aL)l. (15.94) h i h i (D 2)!!Dl − 2 For a stiff chain, the expectation value R will increase aL to aeff L, where aeff is the effective bond length In the limit ofh a veryi large stiffness, called the rod limit, the law (15.94) turns into R2 L2, R2l L2l, (15.95) h i ≡ h i ≡ i.e., the effective bond length length aeff increases to L. This intuitively obvious statement can easily be found from the normalized end-to-end distribution, which coincides in the rod limit with the one-link expression (15.12): 1 rod R PL ( )= D−1 δ(R L), (15.96) SDR − and yields [recall (15.17)] ∞ n D n rod n n R = d R R PL (R)= dR R δ(R L)= L . (15.97) h i Z Z0 − rod By expanding PL (R) in powers of L, we obtain the series 1 ∞ ( 1)n rod R n n n PL ( )= D−1 L R − ∂Rδ(R). (15.98) SDR h i n! nX=0 An expansion of this form holds for any stiffness: the moments of the distribution are the Taylor coefficients of the expansion of PL(R) into a series of derivatives of δ(R)-functions. Let us also calculate the Fourier transformation (15.81) of this distribution. Re- calling (15.15), we find Γ(D/2) P˜rod(q)= P˜rod(qL) J (qL). (15.99) L ≡ (qL/2)D/2−1 D/2−1

For an arbitrary rotationally symmetric PL(R)= PL(R), we simply have to super- impose these distributions for all R: ∞ D−1 rod P˜L(q)= SD dR R P˜ (qL)PL(R). (15.100) Z0

H. Kleinert, PATH INTEGRALS 15.7 Stiﬀ Polymers 1045

This is simply proved by decomposing and performing the Fourier transformation ∞ ′ ′ rod ∞ ′ ′D−1 rod ′ (15.81) on PL(R)= 0 dR δ(R R )PR′ (R)=SD 0 dR R PR′ (R)PL(R ), and − rod performing the FourierR transformation (15.81) on RPR′ (R). In D = 3 dimensions, (15.100) takes the simple form:

∞ 2 sin qR P˜L(q)=4π dR R PL(R). (15.101) Z0 qR Inserting the power series expansion for the Bessel function2 z ν ∞ ( 1)k(z/2)2l J (z)= − (15.102) ν 2 l!Γ(ν + l + 1) Xl=0 into (15.99), and this into (15.100), we obtain

∞ 2l ∞ l q Γ(D/2) D−1 2l P˜L(q)= ( 1) SD dR R R PL(R), (15.103) − 2 l!Γ(D/2+ l) 0 Xl=0 Z in agreement with the general expansion (15.84). The same result is obtained by inserting into (15.103) the expansion (15.98) and using the integrals ∞ dR Rm∂n δ(R)= δ ( 1)nn!, which are proved by n partial integrations. 0 R mn − R The structure factor of a completely stiﬀ polymer (rod limit) is obtained by inserting (15.99) into (15.89). The resulting Srod(q) depends only on qL:

D/2 rod 4 2D 2 2 2 S (qL)= − + Γ(D/2)JD/2−2(qL)+2F (1/2;3/2, D/2; q L /4),(15.104) q2L2 qL! − where F (a; b, c; z) is the hypergeometric function (1.453). For D = 3, the integral (15.89) reduces to (2/L2) L dL′ (L L′)(sin qL′)/qL′, as in the similar equation 0 − (15.9), and the result is simplyR z rod 2 dt S (z)= 2 [cos z 1+ z Si(z)] , Si(z) sin t. (15.105) z − ≡ Z0 t This starts out like 1 z2/36 + z4/1800 + ... . For large z we use the limit of the sine integral3 Si(z) −π/2 to ﬁnd Srod(q) π/qL. → → For of an arbitrary rotationally symmetric end-to-end distribution PL(R), the structure factor can be expressed, by analogy with (15.100), as a superposition of rod limits: ∞ D−1 rod SL(q)= SD dR R S (qR)PL(R). (15.106) Z0 When passing from long to short polymers at a given stiﬀness, there is a crossover between the moments (15.94) and (15.95) and the behaviors of the structure function. Let us study this in detail.

2I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.440. 3ibid., Formulas 8.230 and 8.232. 1046 15 Path Integrals in Polymer Physics

15.7.1 Sliced Path Integral The stiffness of a polymer pictured in Fig. 15.1 may be parameterized by the bending energy κ N EN = (u u )2, (15.107) bend 2a n − n−1 nX=1 where un are the unit vectors specifying the directions of the links. The initial and final link directions of the polymer have a distribution N−1 N 1 dun κ 2 (ubL ua0) = exp (un un−1) , (15.108) | A " A # "−2akBT − # nY=1 Z nX=1 where A is some normalization constant, which we shall choose such that the measure of integration coincides with that of a time-sliced path integral near a unit sphere in Eq. (8.151). Comparison if the bending energy (15.107) with the Euclidean action (8.152) we identify Mr2 κ = , (15.109) hǫ¯ akBT and see that we must replace N N 1 and set → − D−1 A = 2πakBT/κ . (15.110) The result of the integrations in (15.108)q is then known from Eq. (8.156): ∞ ˜ N ∗ κ (ubL ua0) = Il+D/2−1 (h) Ylm(ub)Ylm(ua), h , (15.111) | m ≡ akBT Xl=0 h i X with the modified Bessel function I˜l+D/2−1(z) of Eq. (8.11). The partition function of the polymer is obtained by integrating over all final and averaging over all initial link directions [1]: N N d ua dun κ 2 ZN = exp (un un−1) SD " A # "−2akBT − # Z nY=1 Z nX=1 d ua = d ub (ubL ua0). (15.112) Z Z SD | Inserting here the spectral representation (15.111) we find N κ N 2πκ κ −κ/akB T ZN = I˜D/2−1 = e ID/2−1 . (15.113) akBT "s akBT akBT # Knowing this we may define the normalized distribution function 1 PN (ub, ua)= (ubL ua0), (15.114) ZN | whose integral over ua as well as over ua is unity:

dub PN (ub, ua)= dua PN (ub, ua)=1. (15.115) Z Z

H. Kleinert, PATH INTEGRALS 15.7 Stiﬀ Polymers 1047

15.7.2 Relation to Classical Heisenberg Model The above partition function is closely related to the partition function of the one- dimensional classical Heisenberg model of ferromagnetism which is deﬁned by

N N Heis d ua J ZN dun exp un un−1 , (15.116) ≡ SD "kBT · # Z nY=1 Z nX=1 where J are interaction energies due to exchange integrals of electrons in a ferro- magnet. This diﬀers from (15.112) by a trivial normalization factor, being equal to

2−D N 2πJ J ZHeis = I . (15.117) N  D/2−1  skBT kBT   Identifying J κ/a we may use the Heisenberg partition functions for all cal- ≡ culations of stiﬀ polymers. As an example take the correlation function between neighboring tangent vectors u u . In order to calculate this we observe that h n · n−1i the partition function (15.116) can just as well be calculated exactly with a slight modiﬁcation that the interaction strength J of the Heisenberg model depends on the link n. The result is the corresponding generalization of (15.117):

N 2−D Heis 2πJn Jn ZN (J1,...,JN )= ID/2−1 . (15.118) s kBT kBT  nY=1   This expression may be used as a generating function for expectation values un un−1 which measure the degree of alignment of neighboring spin directions. hIndeed,· wei ﬁnd directly

Heis dZN (J1,...,JN ) ID/2(J/kBT ) un un−1 =(kBT ) = . (15.119) h · i dJn ID/2−1(J/kBT ) Jn≡J

This expectation value measures directly the internal energy per link of te chain. Indeed, since the free energy is F = k T log ZHeis, we obtain [recall (1.548)] N − B N ID/2(J/kBT ) EN = N un un−1 = N . (15.120) h · i ID/2−1(J/kBT ) Let us also calculate the expectation value of the angle between next-to-nearest neighbors u u . We do this by considering the expectation value h n+1 · n−1i 2 Heis 2 2 d ZN (J1,...,JN ) ID/2(J/kBT ) (un+1 un)(un un−1) =(kBT ) = . h · · i dJn+1dJn "ID/2−1(J/kBT )# Jn≡J

(15.121)

Then we prove that the left-hand side is in fact equal to the desired expectation value u u . For this we decompose the last vector u into a component h n+1 · n−1i n+1 1048 15 Path Integrals in Polymer Physics

θn+1,n

u n+1 un θn+1,n−1

θn,n−1 un−1

Figure 15.3 Neighboring links for the calculation of expectation values. parallel to u and a component perpendicular to it: u =(u u ) u +u⊥ . The n n+1 n+1· n n n+1 corresponding decomposition of the expectation value un+1 un−1 is un+1 un−1 = (u u )(u u ) + u⊥ u . Now, the energy inh the Boltzmann· i h factor· dependsi h n+1· n n· n−1 i h n+1· ni only on (un+1 un)+(un un−1) = cos θn+1,n + cos θn,n−1, so that the integral over u⊥ runs over· a surface of· a sphere of radius sin θ in D 1 dimensions [recall n+1 n+1,n − (8.117)] and the Boltzmann factor does not depend on the angles. The integral ⊥ ⊥ receives therefore equal contributions from un+1 and un+1, and vanishes. This proves that − 2 ID/2(J/kBT ) un+1 un−1 = un+1 un un un−1 = , (15.122) h · i h · ih · i "ID/2−1(J/kBT )# and further, by induction, that |l−k| ID/2(J/kBT ) ul uk = . (15.123) h · i "ID/2−1(J/kBT )# For the polymer, this implies an exponential falloﬀ u u = e−|l−k|a/ξ, (15.124) h l · ki where ξ is the persistence length I (κ/ak T ) ξ = a/ log D/2 B . (15.125) − "ID/2−1(κ/akBT )# For D = 3, this is equal to a ξ = . (15.126) −log [coth(κ/ak T ) ak T/κ] B − B Knowing the correlation functions it is easy to calculate the magnetic susceptibility. The total magnetic moment is N M = a un, (15.127) nX=0 so that we ﬁnd the total expectation value 1+ e−a/ξ 1 e−(N+1)a/ξ M2 = a2(N + 1) 2a2e−a/ξ − . (15.128) 1 e−a/ξ − (1 e−a/ξ)2 D E − − The susceptibility is directly proportional to this. For more details see [2].

H. Kleinert, PATH INTEGRALS 15.7 Stiﬀ Polymers 1049

15.7.3 End-to-End Distribution A modiﬁcation of the path integral (15.108) yields the distribution of the end-to-end distance at given initial and ﬁnal directions of the polymer links:

N R = x x = a u (15.129) b − a n nX=1 of the stiﬀ polymer:

N−1 N 1 1 dun (D) PN (ub, ua; R) = δ (R a un) ZN A " A # − nY=2 Z nX=1 N−1 κ 2 exp (un+1 un) , (15.130) × "−2akBT − # nX=1 whose integral over R leads back to the distribution PN (ub, ua):

D d RPN (ub, ua; R)= PN (ub, ua). (15.131) Z If we integrate in (15.130) over all ﬁnal directions and average over the initial ones, we obtain the physically more accessible end-to-end distribution

dua PN (R)= dub PN (ub, ua; R). (15.132) Z Z SD The sliced path integral, as it stands, does not yet give quite the desired probability. Two small corrections are necessary, the same that brought the integral near the surface of a sphere to the path integral on the sphere in Section 8.9. After including these, we obtain a proper overall normalization of PN (ub, ua; R).

15.7.4 Moments of End-to-End Distribution

(D) Since the δ -function in (15.130) contains vectors un of unit length, the calculation of the complete distribution is not straightforward. Moments of the distribution, however, which are deﬁned by the integrals

2l D 2l R = d R R PN (R), (15.133) h i Z are relatively easy to ﬁnd from the multiple integrals

N−1 N 2l 1 D 1 dun dua (D) R = d R dub δ (R aun) h i ZN A " A # " SD # − Z Z nY=2 Z Z nX=1 N−1 2l κ 2 R exp (un+1 un) . (15.134) × "−2akBT − # nX=1 1050 15 Path Integrals in Polymer Physics

Performing the integral over R gives

N−1 N 2l 2l 1 1 dun dua R = dub a un h i ZN A " A # " SD # ! Z nY=2 Z Z nX=1 N−1 κ 2 exp (un+1 un) . (15.135) × "−2akBT − # nX=1 Due to the normalization property (15.115), the trivial moment is equal to unity:

D dua 1 = d RPN (R)= dub PN (ub, ua L)=1. (15.136) h i Z Z Z SD | 15.8 Continuum Formulation

Some properties of stiﬀ polymers are conveniently studied in the continuum limit of the sliced path integral (15.108), in which the bond length a goes to zero and the link number to inﬁnity so that L = Na stays constant. In this limit, the bending energy (15.107) becomes

L κ 2 Ebend = ds (∂su) , (15.137) 2 Z0 where d u(s)= x(s), (15.138) ds is the unit tangent vector of the space curve along which the polymer runs. The parameter s is the arc length of the line elements, i.e., ds = √dx2.

15.8.1 Path Integral If the continuum limit is taken purely formally on the product of integrals (15.108), we obtain a path integral

L −(κ/2k T ) ds [u′(s)]2 (u L u 0) = u e B 0 , (15.139) b | a D Z R This coincides with the Euclidean version of a path integral for a particle on the surface of a sphere. It is a nonlinear σ-model (recall p. 746). The result of the integration has been given in Section 8.9 where we found that it does not quite agree with what we would obtain from the continuum limit of the discrete solution (15.111) using the limiting formula (8.157), which is

∞ kBT ∗ P (ub, ua L)= exp L L2 Ylm(ub)Ylm(ua), (15.140) | − 2κ ! m Xl=0 X

H. Kleinert, PATH INTEGRALS 15.8 Continuum Formulation 1051 with

L =(D/2 1+ l)2 1/4. (15.141) 2 − − For a particle on a sphere, the discrete expression (15.130) requires a correction since it does not contain the proper time-sliced action and measure. According the Section 8.9 the correction replaces L2 by the eigenvalues of the square angular momentum operator in D dimensions, Lˆ2,

L Lˆ2 = l(l + D 2). (15.142) 2 → − After this replacement the expectation of the trivial moment 1 in (15.136) is equal to unity since it gives the distribution (15.140) the proper normalizath i ion:

dub P (ub, ua L)=1. (15.143) Z | This follows from the integral

∗ dub Ylm(ub)Ylm(ua)= δl0 , (15.144) m Z X 2 which was derived in (8.250). Thus, with Lˆ in (15.140) instead of L2, no extra ∗ normalization factor is required. The sum over Ylm(u2)Ylm(u1) may furthermore be rewritten in terms of Gegenbauer polynomials using the addition theorem (8.126), so that we obtain ∞ kBT ˆ2 1 2l + D 2 (D/2−1) P (ub, ua L)= exp L L − Cl (u2u1). (15.145) | − 2κ ! SD D 2 Xl=0 − 15.8.2 Correlation Functions and Moments We are now ready to evaluate the expectation values of R2l. In the continuum approximation we write

L 2l R2l = ds u(s) . (15.146) "Z0 # The expectation value of the lowest moment R2 is given by the double-integral over the correlation function u(s )u(s ) : h i h 2 1 i L L L s2 2 R = ds2 ds1 u(s2)u(s1) =2 ds2 ds1 u(s2)u(s1) . (15.147) h i Z0 Z0 h i Z0 Z0 h i The correlation function is calculated from the path integral via the composition law as in Eq. (3.301), which yields here

dua u(s2)u(s1) = dub du2 du1 h i Z Z SD Z Z P (u , u L s ) u P (u , u s s ) u P (u , u s ).(15.148) × b 2| − 2 2 2 1| 2 − 1 1 1 a| 1 1052 15 Path Integrals in Polymer Physics

The integrals over ua and ub remove the initial and ﬁnal distributions via the normalization integral (15.143), leaving

du1 u(s2)u(s1) = du2 u2u1P (u2, u1 s2 s1). (15.149) h i Z Z SD | −

Due to the manifest rotational invariance, the normalized integral over u1 can be omitted. By inserting the spectral representation (15.145) with the eigenvalues (15.142) we obtain

u(s2)u(s1) = du2 u2u1P (u2, u1 s2 s1) h i Z | − 2 −(s2−s1)kB T Lˆ /2κ 1 2l + D 2 (D/2−1) = e du2 u2u1 − Cl (u2u1) . (15.150) " SD D 2 # Xl Z − We now calculate the integral in the brackets with the help of the recursion relation (15.152) for the Gegenbauer functions4

1 zC(ν)(z)= (2ν + l 1)C(ν) (z)+(l + 1)Cν (z) . (15.151) l 2(ν + l) − l−1 l+1 h i

Obviously, the integral over u2 lets only the term l = 1 survive. This, in turn, involves the integral

(D/2−1) du2 C0 (cos θ)= SD. (15.152) Z The l = 1 -factor D/(D 2) in (15.150) is canceled by the ﬁrst l = 1 -factor in the recursion (15.151) and we− obtain the correlation function

kBT u(s2)u(s1) = exp (s2 s1) (D 1) , (15.153) h i "− − 2κ − # where D 1 in the exponent is the eigenvalue of Lˆ2 = l(l + D 2) at l = 1. The − − correlation function (15.153) agrees with the sliced result (15.124) if we identify the continuous version of the persistence length (15.126) with

ξ 2κ/k T (D 1). (15.154) ≡ B − Indeed, taking the limit a 0 in (15.125), we ﬁnd with the help of the asymptotic behavior (8.12) precisely the→ relation (15.154). After performing the double-integral in (15.147) we arrive at the desired result for the ﬁrst moment:

R2 =2 ξL ξ2 1 e−L/ξ . (15.155) h i − − n h io 4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.933.1.

H. Kleinert, PATH INTEGRALS 15.8 Continuum Formulation 1053

This is valid for all D. The result may be compared with the expectation value of the squared magnetic moment of the Heisenberg chain in Eq. (15.128), which reduces to this in the limit a 0 at ﬁxed L =(N + 1)a. → For small L/ξ, the second moment (15.155) has the large-stiﬀness expansion

1 L 1 L 2 1 L 3 R2 = L2 1 + + ... , (15.156) h i  − 3 ξ 12 ξ ! − 60 ξ !    the first term being characteristic for a completely stiff chain [see Eq. (15.95)]. For large L/ξ, on the other hand, we find the small-stiffness expansion

ξ R2 2ξL 1 + ..., (15.157) h i ≈ − L ! where the dots denote exponentially small terms. The ﬁrst term agrees with relation (15.94) for a random chain with an eﬀective bond length

4 κ a =2ξ = . (15.158) eﬀ D 1 k T − B The calculation of higher expectations R2l becomes rapidly complicated. Take, h i for instance, the moment R4 , which is given by the quadruple integral over the four-point correlation functionh i

L s4 s3 s2 4 R =8 ds4 ds3 ds2 ds1 δi4i3i2i1 ui4 (s4)ui3 (s3)ui2 (s2)ui1 (s1) , (15.159) h i Z0 Z0 Z0 Z0 h i with the symmetric pair contraction tensor of Eq. (15.22):

δ (δ δ + δ δ + δ δ ) . (15.160) i4i3i2i1 ≡ i4i3 i2i1 i4i2 i3i1 i4i1 i3i2 The factor 8 and the symmetrization of the indices arise when bringing the integral

L L L L 4 R = ds4 ds3 ds2 ds1 (u(s4)u(s3))(u(s2)u(s1)) (15.161) Z0 Z0 Z0 Z0 to the s-ordered form in (15.159). This form is needed for the s-ordered evaluation of the u-integrals which proceeds by a direct extension of the previous procedure for R2 . We write down the extension of expression (15.148) and perform the integrals h i over ua and ub which remove the initial and ﬁnal distributions via the normalization integral (15.143), leaving δi4i3i2i1 times an integral [the extension of (15.149)]:

du1 ui4 (s4)ui3 (s3)ui2 (s2)ui1 (s1) = du4 du3 du2 (15.162) h i Z Z Z Z SD u u u u P (u , u s s )P (u , u s s )P (u , u s s ) . × i4 i3 i2 i1 4 3| 4 − 3 3 2| 3 − 2 2 1| 2 − 1 1054 15 Path Integrals in Polymer Physics

The normalized integral over u1 can again be omitted. Still, the expression is complicated. A somewhat tedious calculation yields 4(D + 2) D2 +6D 1 D 7 R4 = L2ξ2 8Lξ3 − − e−L/ξ (15.163) h i D − D2 − D +1 ! D3 + 23D2 7D +1 (D + 5)2 (D 1)5 +4ξ4 − 2 e−L/ξ + − e−2DL/(D−1)ξ . " D3 − (D + 1)2 D3(D + 1)2 # For small values of L/ξ , we find the large-stiffness expansion 2 L 25D 17 L 2 7D2 8D +3 L 3 R4 = L4 1 + − 4 − + ... ,(15.164) h i  − 3 ξ 90(D 1) ξ ! − 315(D 1)2 ξ !  − −   the leading term being equal to (15.95) for a completely stiff chain. In the opposite limit of large L/ξ, the small-stiffness expansion is D +2 D2 +6D 1 ξ D3 + 23D2 7D +1 ξ 2 R4 =4 L2ξ2 1 2 − + − + ..., h i D  − D(D + 2) L D2(D + 2) L!   (15.165) where the dots denote exponentially small terms. The leading term agrees again with the expectation R4 of Eq. (15.94) for a random chain whose distribution is h i (15.49) with an effective link length aeff =2ξ of Eq. (15.158). The remaining terms are corrections caused by the stiffness of the chain. It is possible to find a correction factor to the Gaussian distribution which main- tains the unit normalization and ensures that the moment R2 has the small-ξ exact h i expansion (15.157) whereas R2 is equal to (15.165) up to the first correction term in ξ/L. This has the form h i

D 2 4 D −DR2/4Lξ 2D 1 ξ 3D 1 R D(4D 1) R PL(R)= e 1 − + − − . (15.166) s4πLξ ( − 4 L 4 L2 − 16(D+2) ξL3 ) In three dimensions, this was first written down by Daniels [3]. It is easy to match also the moment R4 by adding in the curly brackets the following terms h i 1 7D+23D2+D3 D +2 ξ2 R2 1 R4 − 1+ + . (15.167) D +1 " 8D L2 ξL ! 32 L4 # These terms do not, however, improve the fits to Monte Carlo data for ξ > 1/10L, since the expansion is strongly divergent. From the approximation (15.166) with the additional term (15.167) we calculate the small-stiffness expansion of all even and odd moments as follows:

n 2 n 2 Γ(D/2+ n/2) n n ξ ξ R = L ξ 1+ A1 + A2 + ... , (15.168) h i Dn/2Γ(D/2)  L L!  where   n 2 2d2 4d (n 1) 1 7d + 23d2 + d3 A = n − − − − , A = n(n 2) − . (15.169) 1 4d (2 + d) 2 − 8d2 (1 + d)

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1055

15.9 Schr¨odinger Equation and Recursive Solution for End-to-End Distribution Moments

The most efficient way of calculating the moments of the end-to-end distribution proceeds by setting up a Schrödinger equation satisfied by (15.112) and solving it recursively with similar methods as developed in 3.19 and Appendix 3C.

15.9.1 Setting up the Schr¨odinger Equation In the continuum limit, we write (15.112) as a path integral [compare (15.139)]

L L ′ D−1 (D) −(¯κ/2) ds [u (s)]2 PL(R) dub dua u δ R ds u(s) e 0 , (15.170) ∝ D − 0 Z Z Z Z ! R where we have introduced the reduced stiﬀness κ ξ κ¯ = =(D 1) , (15.171) kBT − 2 for brevity. After a Fourier representation of the δ-function, this becomes

i∞ dDλ κ¯ ·R/2 PL(R) e dub dua(ubL ua 0) , (15.172) ∝ Z−i∞ 2πi Z Z | where

u(L)=ub L ′ 2 D−1 −(¯κ/2) ds [u (s)] + ·u(s) (ub L ua 0) u e 0 { } (15.173) | ≡ u(0)=ua D Z R describes a point particle of mass M =κ ¯ moving on a unit sphere. In contrast to the

discussion in Section 8.7 there is now an additional external field which prevents us from finding an exact solution. However, all even moments R R R of the h i1 i2 ··· i2l i end-to-end distribution (15.172) can be extracted from the expansion coefficients in powers of λi of the integral dub dua over (15.173). The presence of these directional integrals permits us toR assumeR the external electric field to point in the z-direction, or the Dth direction in D-dimensions. Then = λzˆ, and the moments R2l are proportional to the derivatives (2/κ¯)2l∂2l du du (u L u 0) . λ b a b | a The proportionalityD E factors have been calculated in Eq. (15.84).R ItRis unnecessary to know these since we can always use the rod limit (15.95) to normalize the moments. To find these derivatives, we perform a perturbation expansion of the path integral (15.173) around the solvable case λ = 0. In natural units withκ ¯ = 1, the path integral (15.173) solves obviously the imaginary-time Schrödinger equation

1 1 d

∆u + u + (u τ ua 0) =0, (15.174) −2 2 · dτ ! | 1056 15 Path Integrals in Polymer Physics

where ∆u is the Laplacian on a unit sphere. In the probability distribution (15.211),

only the integrated expression ψ(z, τ; λ) dua(u τ ua 0) (15.175) ≡ Z | appears, which is a function of z = cos θ only, where θ is the angle between u and

the electric ﬁeld . For ψ(z, τ; λ), the Schr¨odinger equation reads d Hˆ ψ(z, τ; λ)= ψ(z, τ; λ), (15.176) −dτ with the simpler Hamiltonian operator 1 Hˆ Hˆ + λ Hˆ = ∆+ λ z ≡ 0 I −2 1 d2 d 1 = (1 z2) (D 1)z + λ z . (15.177) −2 " − dz2 − − dz # 2

Now the desired moments (15.135) can be obtained from the coeﬃcient of λ2l/(2l)! of the power series expansion of the z-integral over (15.175) at imaginary time τ = L:

1 f(L; λ) dz ψ(z, L; λ). (15.178) ≡ Z−1 15.9.2 Recursive Solution of Schr¨odinger Equation. The function f(L; λ) has a spectral representation

∞ 1 (l)† (l) 1 (l) −1 dz ϕ (z) exp E L −1 dza ϕ (za) f(L; λ)= − , (15.179) R 1 dz ϕ(l)†(z) ϕ(l)R(z) Xl=0 −1 R where ϕ(l)(z) are the solutions of the time-independent Schr¨odinger equation Hϕˆ (l)(z) = E(l)ϕ(l)(z). Applying perturbation theory to this problem, we start from the eigenstates of the unperturbed Hamiltonian Hˆ = ∆/2, which are given 0 − by the Gegenbauer polynomials CD/2−1(z) with the eigenvalues E(l) = l(l+D 2)/2. l 0 − Following the methods explained in 3.19 and Appendix 3C we now set up a recursion scheme for the perturbation expansion of the eigenvalues and eigenfunctions [4].Starting point is the expansion of energy eigenvalues and wave-functions in powers of the coupling constant λ:

∞ ∞ (l) (l) (l) j (l) i ′ ′ E = ǫj λ , ϕ = γl′,i λ αl l . (15.180) | i ′ | i jX=0 lX,i=0 The wave functions ϕ(l)(z) are the scalar products z ϕ(l)(λ) . We have inserted h | i extra normalization constants αl′ for convenience which will be ﬁxed soon. The unperturbed state vectors l are normalized to unity, but the state vectors ϕ(l) of | i | i

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1057 the interacting system will be normalized in such a way, that ϕ(l) l = 1 holds to all orders, implying that h | i

(l) (l) γl,i = δi,0 γk,0 = δl,k . (15.181) Inserting the above expansions into the Schr¨odinger equation, projecting the re- j sult onto the base vector k αk, and extracting the coeﬃcient of λ , we obtain the equation h |

∞ i (l) (k) αj (l) (l) (l) γk,iǫ0 + Vk,j γj,i−1 = ǫj γk,i−j , (15.182) αk jX=0 jX=0 where Vk,j = λ k z j are the matrix elements of the interaction between unperturbed states. Forh | i| =i 0, Eq. (15.182) is satisﬁed identically. For i > 0, it leads to the following two recursion relations, one for k = l:

(l) (l) (l) ǫi = γl+n,i−1Wn , (15.183) nX=±1 the other one for k = l: 6 i−1 ǫ(l)γ(l) γ(l) W (l) j k,i−j − k+n,i−1 n (l) jX=1 nX=±1 γk,i = , (15.184) ǫ(k) ǫ(l) 0 − 0 where only n = 1 and n = 1 contribute to the sums over n since − (l) αl+n Wn l z l + n =0, for n = 1. (15.185) ≡ αl h | | i 6 ± The vanishing of W (l) for n = 1 is due to the band-diagonal form of the matrix of n 6 ± the interaction z in the unperturbed basis n . It is this property which makes the sums in (15.183) and (15.184) finite and leads| i to recursion relations with a finite (l) (l) (l) number of terms for all ǫi and γk,i. To calculate Wn , it is convenient to express l z l + n as matrix elements between unnormalized noninteracting states n as h | | i | } l z l + n l z l + n = { | | } , (15.186) h | | i l l l + n l + n { | }{ | } q where expectation values are defined by the integrals

1 D/2−1 D/2−1 2 (D−3)/2 k F (z) l Ck (z) F (z) Cl (z)(1 z ) dz, (15.187) { | | }≡Z−1 − from which we ﬁnd5 24−D Γ(l + D 2) π l l = − . (15.188) { | } l! (2l + D 2) Γ(D/2 1)2 − − 5I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 7.313.1 and 7.313.2. 1058 15 Path Integrals in Polymer Physics

Expanding the numerator of (15.186) with the help of the recursion relation (15.151) for the Gegenbauer polynomials written now in the form

(l + 1) l +1 = (2l + D 2) z l (l + D 3) l 1 , (15.189) | } − | } − − | − } we ﬁnd the only non-vanishing matrix elements to be l +1 l +1 z l = l +1 l +1 , (15.190) { | | } 2l + D 2{ | } l + D −3 l 1 z l = − l 1 l 1 . (15.191) { − | | } 2l + D 2{ − | − } − Inserting these together with (15.188) into (15.186) gives

l(l + D 3) l z l 1 = v − , (15.192) h | | − i u(2l + D 2)(2l + D 4) u − − t and a corresponding result for l z l +1 . We now ﬁx the normalization constants h | | i αl′ by setting

(l) αl+1 W1 = l z l +1 = 1 (15.193) αl h | | i for all l, which determines the ratios

α (l +1) (l + D 2) l = l z l +1 = . (15.194) v − αl+1 h | | i u(2l + D) (2l + D 2) u − t Setting further α1 = 1, we obtain

1/2 l (2l + D 2)(2l + D 4) α = − − . (15.195) l  l(l + D 3)  jY=1 −   Using this we find from (15.185) the remaining nonzero W (l) for n = 1: n − l(l + D 3) W (l) = − . (15.196) −1 (2l + D 2)(2l + D 4) − − (l) We are now ready to solve the recursion relations of (15.183) and (15.184) γk,i and (l) (l) ǫi order by order in i. For the initial order i = 0, the values of the γk,i are (l) given by Eq. (15.181). The coefficients ǫi are equal to the unperturbed energies (l) (l) ǫ0 = E0 = l(l + D 2)/2. For each i = 1, 2, 3,... , there is only a finite number (l) − (l) of non-vanishing γk,j and ǫj with j < i on the right-hand sides of (15.183) and (l) (l) (15.184) which allows us to calculate γk,i and ǫi on the left-hand sides. In this way it is easy to find the perturbation expansions for the energy and the wave functions to high orders.

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1059

Inserting the resulting expansions (15.180) into Eq. (15.179), only the totally symmetric parts in ϕ(l)(z) will survive the integration in the numerators, i.e., we may insert only

∞ ϕ(l) (z)= z ϕ(l) = γ(l) λi z 0 . (15.197) symm h | symmi 0,i h | i Xi=0 (l) ′ 2 2i The denominators of (15.179) become explicitly l′,i γl′,i αl λ , where the sum 2 | | over i is limited by power of λ up to which weP want to carry the perturbation series; also l′ is restricted to a finite number of terms only, because of the band- (l) diagonal structure of the γl′,i. Extracting the coefficients of the power expansion in λ from (15.179) we obtain all desired moments of the end-to-end distribution, in particular the second and fourth moments (15.155) and (15.164). Higher even moments are easily found with the help of a Mathematica program, which is available for download in notebook form [5]. The expressions are too lengthy to be written down here. We may, however, expand the even moments Rn in powers of L/ξ to find a general large-stiffness h i expansion valid for all even and odd n: Rn n L n ( 13 n +5D (1 + n)) L2 L3 L4 h i =1 + − − a + a +..., (15.198) Ln − 6 ξ 360(D 1) ξ2 − 3 ξ3 4 ξ4 − where 444 63n+15n2+7D2 (4+15n+5n2)+2D ( 124 141n+7n2) a3 = n − − − , 45360(D 1)2 − n 2 3 a4 = D0 + D1d + D2d + D3d , (15.199) 5443200(d 1)3 − with

D =3 5610+2921n 822n2+67n3 ,D = 8490+12103n 3426n2+461n3, 0 − − 1 − D =45 2 187n 46n2 +7n3 ,D = 35 6+31n+30n2+5n3 . (15.200) 2 − − − 4 − The lowest odd moments are, up to order l4,

R l 5D 7 33 43D + 14D2 861 1469D + 855D2 175D3 h i = 1 + − l2 − l3 − − l4 ..., L − 6 180(D 1) − 3780(D 1)2 − 453600 (D 1)3 − − − R3 l 5D 4 195 484D+329D2 609 2201D+2955D2 1435D3 = 1 + − l2 − l3 − − l4 .... L3 − 2 30 (D 1) − 7560 ( 1+ d)2 − 151200 (D 1)3 − − − 15.9.3 From Moments to End-to-End Distribution for D=3 We now use the recursively calculated moments to calculate the end-to-end distribution itself. It can be parameterized by an analytic function of r = R/L [4]:

P (R) rk(1 rβ)m, (15.201) L ∝ − 1060 15 Path Integrals in Polymer Physics whose moments are exactly calculable:

3+ k +2 l 3+ k Γ Γ + m +1 β β r2l = . (15.202) 3+ k +2 l Γ 3+k Γ + m +1 β β We now adjust the three parameters k, β, and m to fit the three most important moments of this distribution to the exact values, ignoring all others. If the distances were distributed uniformly over the interval r [0, 1], the moments would ∈ be r2l unif = 1/(2l + 2). Comparing our exact moments r2l (ξ) with those of h i the uniform distribution we find that r2l (ξ)/ r2l unif has a maximumD E for n close to h i h i nmax(ξ) 4ξ/L. We identify the most important moments as those with n = nmax(ξ) and n =≡n (ξ) 1. If n (ξ) 1, we choose the lowest even moments r2 , r4 , max ± max ≤ h i h i and r6 . In particular, we have fitted r2 , r4 and r6 for small persistence length ξ

17.5 k 80 22 15 β 20 12.5 660 18 10 m 40 7.5 16 14 5 20 2.5 12 0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2 ξ/L ξ/L ξ/L

Figure 15.4 Paramters k, β, and m for a best ﬁt of end-to-end distribution (15.201).

For small persistence lengths ξ/L =1/400, 1/100, 1/30, the curves are well approximated by Gaussian random chain distributions on a lattice with lattice constant −3R2/4Lξ aeff =2ξ, i.e., PL(R) e [recall (15.75)]. This ensures that the lowest mo- 2 → ment R = aeff L is properly fitted. In fact, we can easily check that our fitting programh yieldsi for the parameters k,β,m in the end-to-end distribution (15.201) the ξ 0 behavior: k ξ, β 2+2ξ, m 3/4ξ, so that (15.201) tends to the correct→ Gaussian behavior.→ − → → In the opposite limit of large ξ, we find that k 10ξ 7/2, β 40ξ+5, m 10, → − → → which has no obvious analytic approach to the exact limiting behavior PL(R) (1 r)−5/2e−1/4ξ(1−r), although the distribution at ξ = 2 is fitted numerically extremely→ − well.

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1061

The distribution functions can be inserted into Eq. (15.89) to calculate the structure factors shown in Fig. 15.5. They interpolate smoothly between the Debye limit (15.90) and the stiﬀ limit (15.105). 1 0.8 ξ/L =2 0.6 ✠ ξ/L =1 S(q) ξ/L =1/2 0.4 ✠ ✠ ξ/L =1/5,..., 1/400 0.2 ✠

q√ξ 5 10 15 20 25 30

Figure 15.5 Structure functions for different persistence lengths ξ/L = 1/400, 1/100, 1/30, 1/10, 1/5, 1/2, 1, 2, (from bottom to top) following from the end-to-end distributions in Fig. 15.6. The curves with low ξ almost coincide in this plot over the ξ-dependent absissa. The very stiff curves fall off like 1/q, the soft ones like 1/q2 [see Eqs. (15.105) and (15.90)].

15.9.4 Large-Stiffness Approximation to End-to-End Distribution The full end-to-end distribution (15.132) cannot be calculated exactly. It is, however, quite easy to find a satisfactory approximation for large stiffness [6]. We start with the expression (15.170) for the end-to-end distribution PL(R). In Eq. (3.233) we have shown that a harmonic path integral including the integrals over the end points can be found, up to a trivial factor, by summing over all paths with Neumann boundary conditions. These are satisfied if we expand the fields u(s) into a Fourier series of the form (2.452):

∞

u(s)= u0 + (s)= u0 + un cos νns, νn = nπ/L. (15.203) nX=1 Let us parametrize the unit vectors u in D dimensions in terms of the ﬁrst D 1 -dimensional coordinates uµ qµ with µ = 1,...,D 1. The Dth component− is ≡ − then given by a power series

σ 1 q2 1 q2/2 (q2)2/8+ .... (15.204) ≡ − ≈ − − q The we approximate the action harmonically as follows: κ¯ L 1 = (0) + int = ds [u′(s)]2 + δ(0) log(1 q2) A A A 2 Z0 2 − κ¯ L 1 L ds [q′(s)]2 δ(0) ds q2. (15.205) ≈ 2 Z0 − 2 Z0 1062 15 Path Integrals in Polymer Physics

The last term comes from the invariant measure of integration dD−1q/√1 q2 [recall (10.636) and (10.641)]. − Assuming, as before, that R points into the z, or Dth, direction we factorize L L 1 1 δ(D) R ds u(s) = δ R L + ds q2(s)+ [q2(s)]2 + ... −Z0 ! − Z0 2 8 ! L δ(D−1) ds q(s) , (15.206) × Z0 ! where R R . The second δ-function on the right-hand side enforces ≡| | L q¯ = L−1 ds qµ(s)=0 , µ =1,...,d 1 , (15.207) Z0 − µ and thus the vanishing of the zero-frequency parts q0 in the ﬁrst D 1 components of the Fourier decomposition (15.203). − It was shown in Eqs. (10.632) and (10.642) that the last δ-function has a dis- torting eﬀect upon the measure of path integration which must be compensated by a Faddeev-Popov action

L FP D 1 2 e = − ds q , (15.208) A 2L Z0 where the number D of dimensions of qµ-space (10.642) has been replaced by the present number D 1. In the large-stiﬀness− limit we have to take only the ﬁrst harmonic term in the action (15.205) into account, so that the path integral (15.170) becomes simply

L L ′ ′ D−1 1 2 −(¯κ/2) ds [q (s)]2 PL(R) q δ R L + ds q (s) e 0 . (15.209) ∝ NBC D − 0 2 Z Z ! R The subscript of the integral emphasizes the Neumann boundary conditions. The prime on the measure of the path integral indicates the absence of the zero-frequency component of qµ(s) in the Fourier decomposition due to (15.207). Representing the remaining δ-function in (15.209) by a Fourier integral, we obtain

i∞ 2 L dω κω¯ 2(L−R) ′ D−1 κ¯ ′2 2 2 PL(R) κ¯ e q exp ds q + ω q . (15.210) ∝ −i∞ 2πi NBC D "− 2 0 # Z Z Z The integral over all paths with Neumann boundary conditions is known from Eq. (2.456). At zero average path, the result is

κ¯ L ωL (D−1)/2 ′ D−1q exp ds q′2 + ω2q2 , (15.211) NBC D "− 2 0 # ∝ sinh ωL Z Z so that

i∞ 2 (D−1)/2 dω κω¯ 2(L−R) ωL PL(R) κ¯ e . (15.212) ∝ Z−i∞ 2πi sinh ωL

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1063

The integral can easily be done in D = 3 dimensions using the original product representation (2.455) of ωL/ sinh ωL, where

i∞ 2 ∞ 2 −1 dω 2 ω R κω¯ (L−R) PL( ) κ¯ e 1+ 2 . (15.213) ∝ −i∞ 2πi νn ! Z nY=1 If we shift the contour of integration to the left we run through poles at ω2 = ν2 − k with residues

∞ ∞ 2 −1 2 k νk 1 . (15.214) − n2 ! kX=1 n(6=Yk)=1 The product is evaluated by the limit procedure for small ǫ:

∞ (k + ǫ)2 −1 (k + ǫ)2 (k + ǫ)π 2ǫ 2ǫπ 1 1 − −  " − n2 #  " − k2 # → sin(k + ǫ)π k → sin(k + ǫ)π nY=1  2ǫπ   2( 1)k+1. (15.215) → −cos kπ sin ǫπ → − Hence we obtain [6]

∞ k+1 2 −κν¯ 2(L−R) P (R) 2( 1) κ¯ ν e k . (15.216) L ∝ − k kX=1 It is now convenient to introduce the reduced end-to-end distance r R/L and the 2 2 2 ≡ ﬂexibility of the polymer l L/ξ, and replaceκ ¯ νk (L R) k π (1 r)/l so that (15.216) becomes ≡ − → −

∞ 2 2 P (R)= L(R)= ( 1)k+1k2π2e−k π (1−r)/l, (15.217) L N N − kX=1 where is a normalization factor determined to satisfy N ∞ 3 3 2 d RPL(R)=4πL drr PL(R)=1. (15.218) Z Z0 The sum must be evaluated numerically, and leads to the distributions shown in Fig. 15.6. The above method is inconvenient if D = 3, since the simple pole structure of (15.213) is no longer there. For general D, we6 expand

ωL (D−1)/2 ∞ (D 1)/2 =(ωL)(D−1)/2 ( 1)k − − e−(2k+(D−1)/2)ωL, (15.219) sinh ωL − k ! kX=0 and obtain from (15.212):

∞ k (D 1)/2 PL(R) ( 1) − − Ik(R/L), (15.220) ∝ − k ! kX=0 1064 15 Path Integrals in Polymer Physics

2 4πr PL(R) ξ/L=

Figure 15.6 Normalized end-to-end distribution of stiff polymer according to our analytic formula (15.201) plotted for persistence lengths ξ/L = 1/400, 1/100, 1/30, 1/10, 1/5, 1/2, 1, 2 (fat curves). They are compared with the Monte Carlo calculations (symbols) and with the large-stiffness approximation (15.217) (thin curves) of Ref. [6] which fits well for ξ/L = 2 and 1 but becomes bad small ξ/L < 1. For very small values such as ξ/L = 1/400, 1/100, 1/30, our theoretical curves are well approximated by Gaussian random chain distributions on a lattice with lattice constant a = 2ξ of Eq. (15.158) which ensures that the lowest moments R2 = a L are properly eff h i eff fitted. The Daniels approximation (15.166) fits our theoretical curves well up to larger ξ/L 1/10. ≈ with the integrals

i∞ dω¯ (D+1)/2 −[2k+(D−1)/2]¯ω+(D−1)¯ω2(1−r)/2l Ik(r) ω¯ e , (15.221) ≡ Z−i∞ 2πi whereω ¯ is the dimensionless variable ωL. The integrals are evaluated with the help of the formula6

i∞ dx ν βx2/2−qx 1 −q2/4β x e = e Dν q/ β , (15.222) −i∞ 2πi √ (ν+1)/2 Z 2πβ q where Dν(z) is the parabolic cylinder functions which for integer ν are proportional to Hermite polynomials:7

1 −z2/4 Dn(z)= n e Hn(z/√2). (15.223) √2 Thus we ﬁnd 6I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 3.462.3 and 3.462.4. 7ibid., Formula 9.253.

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1065

(D+3)/4 1 l [2k+(D−1)/2]2 2k+(D 1)/2 − 4(D−1)(1−r)/l Ik(r)= e D(D+1)/2 − , √2π " (D 1)(1 r)#  (D 1)(1 r)/l  − − − − q (15.224) which becomes for D =3

2 1 1 − (2k+1) 2k+1 I (r)= e 4(1−r)/l H . (15.225) k 3 2  2√2π 2(1 r)/l 2 (1 r)/l − − q  q  If the sum (15.220) is performed numerically for D = 3, and the integral over PL(R) is normalized to satisfy (15.218), the resulting curves fall on top of those in Fig. 15.6 which were calculated from (15.217). In contrast to (15.217), which converges rapidly for small r, the sum (15.220) convergent rapidly for r close to unity. Let us compare the low moments of the above distribution with the exact moments in Eqs. (15.156) and (15.164). in the large-stiffness expansion. We setω ¯ ωL ≡ and expand D−1 ω¯ f(¯ω2) (15.226) ≡ ssinhω ¯ in a power series D 1 (D 1)(5D 1) (D 1)(15+14D+35D2) f(¯ω2)=1 − ω¯ 2 + − − ω¯4 − ω¯6 +....(15.227) − 22 3 25 32 5 − 27 34 5 7 · · · · · · Under the integral (15.212), each power ofω ¯ 2 may be replaced by a differential operator L d 2l d ω¯ 2 ω¯ˆ2 = . (15.228) → ≡ − κ¯ dr −D 1 dr − The expansion f(ω¯ˆ2) can then be pulled out of the integral, which by itself yields a δ-function, so that we obtain P (R) in the form f ω¯ˆ2 δ(r 1), which is a series of L − derivatives of δ-functions of r 1 staring with − l d ( 1+5 D) l2 d2 (15+14 D+35 D2) l3 d3 PL(R) 1+ + − + 2 + ... δ(r 1). ∝ " 6 dr 360 (D 1) dr2 45360 (D 1) dr3 # − − − (15.229) From this we find easily the moments ∞ m D m D−1 m R = d RR PL(R) drr r PL(R). (15.230) h i Z ∝ Z0 Let us introduce auxiliary expectation values with respect to the simple integrals f(r) 1 dr f(r)PL, rather than to D-dimensional volume integrals. The unnor- malizedh i ∝ moments rm are then given by rD−1+m . Within these one-dimensional R h i h i1 expectations, the moments of z = r 1 are − ∞ n n (r 1) 1 dr (r 1) PL(R). (15.231) h − i ∝ Z0 − 1066 15 Path Integrals in Polymer Physics

m m D−1+m The moments r [1+(r 1)] = [1+(r 1)] 1 are obtained by expanding h ih − i h − i m (n) m the binomial in powers of r 1 and using the integrals dz z δ (z)=( 1) m! δmn to ﬁnd, up to the third power− in L/ξ = l, − R D 1 (5D 1)(D 2) (35D3 +14D+15)(D 2)(D 3) R0 = 1 − l + − − l2 − − l3 , N − 6 360 − 45360(D 1) − D +1 (5D 1)D(D+1) (35D2 +14D+ 15)D(D+1) R2 = L2 1 l + − l2 l3 , N − 6 360(D 1) − 45360(D 1) − − D +3 (5D 1)(D+2)(D+3) R4 = L4 1 l + − l2 N − 6 360(D 1) − (35D2 +14D+15)(D+1)(D+2)(D+3) l3 . − 45360(D 1)2 − The zeroth moment determines the normalization factor to ensure that R0 = 1. Dividing this out of the other moments yields N h i

1 13D 9 8 R2 = L2 1 l + − l2 l3 + ... , (15.232) " − 3 180(D 1) − 945 # D E − 2 23D 11 123D2 98D+39 R4 = L4 1 l + − l2 − l3+ ... . (15.233) " − 3 90(D 1) − 1890(D 1)2 # D E − − These agree up to the l-terms with the exact expansions (15.156) and (15.164) [or with the general formula (15.198)]. For D = 3, these expansions become 1 1 8 R2 = L2 1 l + l2 l3 + ... , (15.234) − 3 12 − 945 D E 2 29 71 R4 = L4 1 l + l2 l3 + ... . (15.235) − 3 90 − 630 D E Remarkably, these happen to agree in one more term with the exact expansions (15.156) and (15.164) than expected, as will be understood after Eq. (15.295).

15.9.5 Higher Loop Corrections Let us calculate perturbative corrections to the large-stiﬀness limit. For this we replace the harmonic path integral (15.210) by the full expression

L tot cor −1 ′ D−1 −1 2 −A [q]−A [qb,qa] P (r; L)=SD q(s)δ r L ds 1 q (s) e , (15.236) NBCD − 0 − ! Z Z p where 1 L R [q]= ds [ g (q)q ˙µ(s)q ˙ν (s) εδ(s,s)log g(q(s))] + FP εL . (15.237) Atot 2ε µν − A − 8 Z0

For convenience, we have introduced here the parameter ε = kBT/κ = 1/κ¯, the reduced inverse stiﬀness, related to the ﬂexibility l L/ξ by l = εL(d 1)/2. We also have added the correction term εLR/8 to have a unit normalization≡ of the partition− function

∞ D−1 Z = SD dr r P (r; L)=1. (15.238) Z0

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1067

The Faddeev-Popov action, whose harmonic approximation was used in (15.208), is now

L FP[q]= (D 1)log L−1 ds 1 q2(s) , (15.239) A − − 0 − ! Z p To perform higher-order calculations we have added an extra action which corrects for the omission of ﬂuctuations of the velocities at the endpoints when restricting the paths to Neumann boundary conditions with zero end-point velocities. The extra action contains, of course, only at the endpoints and reads [7]

cor[q , q ]= log J[q , q ]= [q2(0) + q2(L)]/4. (15.240) A b a − b a − Thus we represent the partition function (15.238) by the path integral with Neumann boundary conditions

Z = ′ D−1q(s)exp [q] cor[q , q ] FP[q] . (15.241) D −Atot − A b a − A ZNBC A similar path integral can be set up for the moments of the distribution. We express the square distance R2 in coordinates (15.207) as

2 L L L R2 = ds ds′ u(s) u(s′)= ds 1 q2(s) = R2 , (15.242) 0 0 · 0 − ! Z Z Z p we ﬁnd immediately the following representation for all, even and odd, moments [compare (15.147)]

n L L (R2)n = ds ds′ u(s) u(s′) (15.243) h i · * "Z0 Z0 # + in the form

Rn = ′ D−1q(s)exp [q] [q , q ] FP[q] . (15.244) h i D −Atot − Acor b a − An ZNBC This diﬀers from Eq. (15.241) only by in the Faddeev-Popov action for these moments, which is

L FP −1 2 n [q]= (n + D 1)log L ds 1 q (s) , (15.245) A − − 0 − ! Z p rather than (15.239). There is no need to divide the path integral (15.244) by Z since this has unit normalization, as will be veriﬁed order by order in the perturbation expansion. The Green function of the operator d2/ds2 with these boundary conditions has the form

L s s′ (s + s′) (s2 + s′2) ∆′ (s,s′)= | − | + . (15.246) N 3 − 2 − 2 2L The zero temporal average (15.207) manifests itself in the property

L ′ ′ ds ∆N(s,s ) = 0 . (15.247) Z0 ′ ′ ′ In the following we shall simply write ∆(s,s ) for ∆N(s,s ), for brevity. 1068 15 Path Integrals in Polymer Physics

Partition Function and Moments Up to Four Loops We are now prepared to perform the explicit perturbative calculation of the partition function and all even moments in powers of the inverse stiﬀness ε up to order ε2 l2. This requires evaluating Feynman diagrams up to four loops. The associated integrals will contain∝ products of distributions, which will be calculated unambiguously with the help of our simple formulas in Chapter 10. For a systematic treatment of the expansion parameter ε, we rescale the coordinates qµ εqµ, and rewrite the path integral (15.244) as →

Rn = ′ D−1q(s)exp [q; ε]. , (15.248) h i D {−Atot,n } ZNBC with the total action L 1 (qq˙)2 1 [q; ε] = ds q˙2 + ε + δ(0) log(1 εq2) Atot,n 2 1 εq2 2 − Z0 − 1 L ε R σ log ds 1 εq2 q2(0) + q2(L) εL . (15.249) − n L − − 4 − 8 " Z0 # p The constant σn is an abbreviation for σ n + (d 1). (15.250) n ≡ − For n = 0, the path integral (15.248) must yield the normalized partition function Z = 1. For the perturbation expansion we separate

[q; ε]= (0)[q]+ int[q; ε], (15.251) Atot,n A An with a free action 1 L (0)[q]= ds q˙2(s), (15.252) A 2 Z0 and a large-stiﬀness expansion of the interaction

int[q; ε]= ε int1[q]+ ε2 int2[q]+ .... (15.253) An An An The free part of the path integral (15.248) is normalized to unity:

L 2 (0) ′ D−1 −A(0) [q] ′ D−1 −(1/2) ds q˙ (s) Z q(s) e = q(s) e 0 =1. (15.254) ≡ D D ZNBC ZNBC R The ﬁrst expansion term of the interaction (15.253) is

1 L R int1[q] = ds [q(s)q ˙(s)]2 ρ (s) q2(s) L , (15.255) An 2 − n − 8 Z0 where

ρ (s) δ + [δ(s)+ δ(s L)] /2, δ δ(0) σ /L. (15.256) n ≡ n − n ≡ − n The second term in ρn(s) represents the end point terms in the action (15.249) and is important for canceling singularities in the expansion. The next expansion term in (15.253) reads

L int2 1 2 1 σn 2 2 n [q] = ds [q(s)q ˙(s)] δ(0) q (s) q (s) A 2 0 − 2 − 2L Z h i σ L L + n ds ds′ q2(s)q2(s′). (15.257) 8L2 Z0 Z0

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1069

The perturbation expansion of the partition function in powers of ε consists of expectation values of the interaction and its powers to be calculated with the free partition function (15.254). For an arbitrary functional of q(s), these expectation values will be denoted by

L 2 ′ D−1 −(1/2) ds q˙ (s) F [q] q(s) F [q] e 0 . (15.258) h i0 ≡ D ZNBC R With this notation, the perturbative expansion of the path integral (15.248) reads

1 Rn /Ln = 1 int[q; ε] + int[q; ε]2 ... h i − hAn i0 2hAn i0 − 1 = 1 ε int1[q] + ε2 int2[q] + int1[q]2 .... (15.259) − hAn i0 −hAn i0 2hAn i0 − For the evaluation of the expectation values we must perform all possible Wick contractions with the basic propagator

qµ(s)qν (s′) = δµν ∆(s,s′) , (15.260) h i0 where ∆(s,s′) is the Green function (15.246) of the unperturbed action (15.252). The relevant loop integrals Ii and Hi are calculated using the dimensional regularization rules of Chapter 10. They are listed in Appendix 15A and Appendix 15B. We now state the results for various terms in the expansion (15.259):

(D 1) σ 1 1 R (D 1)n int1[q] = − n I + DI ∆(0, 0) ∆(L,L) L = L − , hAn i0 2 L 1 2 − 2 − 2 − 8 12 (D2 1) σ (D 1)σ int2[q] = − δ(0) + n I +2(D+2)I + − n (D 1)I2 +2I hAn i0 4 2L 3 4 8L2 − 1 5 (D2 h1) (D 1) i = L3 − δ(0) + L2 − (25D2 +36D+23)+ n(11D+5) , 120 1440 1 L2 (D 1)L D 2 R 2 int1[q]2 = − δ(0) + δ + 2hAn i0 2 12 2L − n L − 8 (D 1) + − [Hn 2(Hn + Hn H )+ H 4D(Hn H H ) 4 1 − 2 3 − 5 6 − 4 − 7 − 10 (D 1) + H +2D2(H + H ) + − [DH + 2(D + 2)H + DH ] 11 8 9 4 12 13 14 L2 (D 1)n 2 (D 1) = − + L3 − δ(0) 2 12 120 (D 1) + L2 − (25D2 22D +25)+4(n +4D 2) 1440 − − (D 1)D (D 1) + L3 − δ(0) + L2 − (29D 1) . (15.261) 120 720 − Inserting these results into Eq. (15.259), we ﬁnd all even and odd moments up to order ε2 l2 ∝ (D 1)n (D 1)2n2 (D 1)(4n +5D 13)n Rn /Ln = 1 εL − + ε2L2 − + − − (ε3) h i − 12 288 1440 − O n n2 (4n +5D 13)n = 1 l + + − l2 (l3). (15.262) − 6 72 360(D 1) − O − For n = 0 this gives the properly normalized partition function Z = 1. For all n it reproduces the large-stiﬀness expansion (15.198) up to order l4. 1070 15 Path Integrals in Polymer Physics

Correlation Function Up to Four Loops As an important test of the correctness of our perturbation theory we calculating the correlation function up to four loops and verify that it yields the simple expression (15.153), which reads in the present units

′ ′ G(s,s′)= e−|s−s |/ξ = e−|s−s |l/L. (15.263)

Starting point is the path integral representation with Neumann boundary conditions for the two- point correlation function

′ ′ ′ D−1 ′ (0) G(s,s )= u(s) u(s ) = q(s) f(s,s ) exp tot[q; ε] , (15.264) h · i NBC D −A Z n o with the action of Eq. (15.249) for n = 0. The function in the integrand f(s,s′) f(q(s), q(s′)) is an abbreviation for the scalar product u(s) u(s′) expressed in terms of independent≡ coordinates qµ(s): ·

f(q(s), q(s′)) u(s) u(s′)= 1 q2(s) 1 q2(s′)+ q(s) q(s′). (15.265) ≡ · − − Rescaling the coordinates q √ε q, and expandingp in powersp of ε yields: → ′ ′ 2 ′ f(q(s), q(s ))=1+ εf1(q(s), q(s )) + ε f2(q(s), q(s )) + ..., (15.266) where 1 1 f (q(s), q(s′)) = q(s) q(s′) q2(s) q2(s′), (15.267) 1 − 2 − 2 1 1 1 f (q(s), q(s′)) = q2(s) q2(s′) [q2(s)]2 [q2(s′)]2. (15.268) 2 4 − 8 − 8 We shall attribute the integrand f(q(s), q(s′)) to an interaction f [q; ε] deﬁned by A f f(q(s), q(s′)) e−A [q;ε], (15.269) ≡ which has the ε-expansion

f [q; ε] = log f(q(s), q(s′)) A − 1 = εf (q(s), q(s′))+ε2 f (q(s), q(s′))+ f 2(s,s′) ..., (15.270) − 1 − 2 2 1 − to be added to the interaction (15.253) with n = 0. Thus we obtain the perturbation expansion of the path integral (15.264)

′ int f 1 int f 2 G(s,s )=1 ( 0 [q; ε]+ [q; ε]) + ( 0 [q; ε]+ [q; ε]) .... (15.271) − A A 0 2 A A 0 − D E D E Inserting the interaction terms (15.253) and (15.271), we obtain

G(s,s′)=1+ε f (q(s), q(s′)) +ε2 f (q(s), q(s′)) f (q(s), q(s′)) int1[q] +..., h 1 i0 h 2 i0 −h 1 A0 i0 (15.272) and the expectation values can now be calculated using the propagator (15.260) with the Green function (15.246). In going through this calculation we observe that because of translational invariance in the ′ pseudotime, s s + s0, the Green function ∆(s,s )+ C is just as good a Green function satisfying Neumann boundary→ conditions as ∆(s,s′). We may demonstrate this explicitly by setting C =

H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1071

L(a 1)/3 with an arbitrary constant a, and calculating the expectation values in Eq. (15.272) using the− modiﬁed Green function. Details are given in Appendix 15C [see Eq. (15C.1)], where we list various expressions and integrals appearing in the Wick contractions of the expansion Eq. (15.272). Using these results we ﬁnd the a-independent terms up to second order in ε:

(D 1) f (q(s), q(s′)) = − s s′ , (15.273) h 1 i0 − 2 | − | f (q(s), q(s′)) f (q(s), q(s′)) int1[q] h 2 i0 − h 1 A0 i0 1 1 (D 1) 1 1 = (D 1) D (D + 1)D2 K DK − K + K + K − 2 1 − 8 2 − 1 − 2 − L 3 2 4 2 5 1 = (D 1)2(s s′)2. (15.274) 8 − − This leads indeed to the correct large-stiﬀness expansion of the exact two-point correlation function (15.263):

D 1 (D 1)2 s s′ (s s′)2 G(s,s′) = 1 ε − s s′ + ε2 − (s s′)2 +...=1 | − | + − .... (15.275) − 2 | − | 8 − − ξ 2ξ2 −

Radial Distribution up to Four Loops We now turn to the most important quantity characterizing a polymer, the radial distribution function. We eliminate the δ-function in Eq. (15.236) enforcing the end-to-end distance by considering the Fourier transform

P (k; L)= dr eik(r−1) P (r; L) . (15.276) Z This is calculated from the path integral with Neumann boundary conditions

P (k; L)= ′ D−1q(s)exp tot[q; ε] , (15.277) D −Ak ZNBC where the action tot[q; ε] reads, with the same rescaled coordinates as in (15.249), Ak L 1 (qq˙)2 1 ik tot[q; ε] = ds q˙2 + ε + δ(0) log(1 εq2) 1 εq2 1 Ak 2 1 εq2 2 − − L − − Z0 − 1 R p ε q2(0) + q2(L) εL 0[q]+ int[q; ε]. (15.278) − 4 − 8 ≡ A Ak As before in Eq. (15.253), we expand the interaction in powers of the coupling constant ε. The ﬁrst term coincides with Eq. (15.255), except that σn is replaced by

ρ (s)= δ + [δ(s)+ δ(s L)] /2, δ = δ(0) ik/L, (15.279) k k − k − so that

L int1 1 2 2 R k [q] = ds [q(s)q ˙(s)] ρk(s) q (s) L . (15.280) A 0 2 − − 8 Z n o int2 The second expansion term k [q] is simpler than the previous (15.257) by not containing the last nonlocal term: A

L 1 1 ik int2[q]= ds [q(s)q ˙(s)]2 δ(0) q2(s) q2(s). (15.281) Ak 2 − 2 − 2L Z0 1072 15 Path Integrals in Polymer Physics

Apart from that, the perturbation expansion of (15.277) has the same general form as in (15.259):

1 P (k; L)=1 ε int1[q] + ε2 int2[q] + int1[q]2 .... (15.282) − hAk i0 −hAk i0 2hAk i0 − The expectation values can be expressed in terms of the same integrals listed in Appendix 15A and Appendix 15B as follows:

(D 1) ik 1 1 R int1[q] = − I + DI ∆(0, 0) ∆(L,L) L hA, k i0 2 L 1 2 − 2 − 2 − 8 (D 1) [(D 1) ik] = L − − − , (15.283) − 12 (D2 1) ik int2[q] = − δ(0) + I + 2(D + 2)I hA, k i0 4 2L 3 4 (D2 1) (D2 1) [7(D +2)+3ik] = L3 − δ(0) + L2 − , (15.284) 120 720 1 L2 (D 1)L D 2 R 2 int1[q]2 = − δ(0) + δ + 2hA, k i0 2 12 2L − k L − 8 (D 1) + − Hk 2(Hk + Hk H )+ H 4D(Hk H H ) 4 1 − 2 3 − 5 6 − 4 − 7 − 10 (D 1) + H +2D2(H + H ) + − [DH + 2(D + 2)H + DH ] 11 8 9 4 12 13 14 (D 1)2 [(D 1) ik]2 = L2 − − − 2 122 (D 1) · (D 1) + L3 − δ(0) + L2 − (13D2 6D +21)+4ik(2D + ik) 120 1440 − (D 1)D (D 1) + L3 − δ(0) + L2 − (29D 1) . (15.285) 120 720 − In this way we ﬁnd the large-stiﬀness expansion up to order ε2:

(D 1) (D 1) P (k; L)=1+ εL − [(D 1) ik]+ ε2L2 − (15.286) 12 − − 1440 (ik)2(5D 1) 2ik(5D2 11D+8)+(D 1)(5D2 11D+14) + (ε3). × − − − − − O This can also be rewritten as

(D 1) (D 3) (5D2 11D+14) P (k; L)=P (k; L) 1+ − l+ − ik+ − l2 + (l3) , 1 loop 6 180(D 1) 360 O − (15.287) where the prefactor P1 loop(k; L) has the expansion

(D 1) (D 1)(5D 1) P (k; L)=1 εL − (ik)+ ε2L2 − − (ik)2 .... (15.288) 1 loop − 22 3 25 32 5 − · · · With the identiﬁcationω ¯2 = ikεL, this is the expansion of one-loop functional determinant in (15.227). By Fourier-transforming (15.286), we obtain the radial distribution function

l l2 P (r; l) = δ(r 1)+ [δ′(r 1)+(d 1) δ(r 1)] + [(5d 1) δ′′(r 1) − 6 − − − 360(d 1) − − − + 2(5d2 11d+8) δ′(r 1)+(d 1)(5d2 11d+14)δ(r 1) + (l3). (15.289) − − − − − O H. Kleinert, PATH INTEGRALS 15.9 Schr¨odinger Equation and Recursive Solution for Moments 1073

As a crosscheck, we can calculate from this expansion once more the even and odd moments

Rn = Ln dr rn+(D−1) P (r; l), (15.290) h i Z and ﬁnd that they agree with Eq. (15.262). Using the higher-order expansion of the moments in (15.198) we can easily extend the distribution (15.289) to arbitrarily high orders in l. Keeping only the terms up to order l4, we ﬁnd that the one-loop end-to-end distribution function (15.212) receives a correction factor:

∞ 2 (D−1)/2 dωˆ 2 ω¯ 2 P (r; l) e−iωˆ (r−1)(D−1)/2l e−V (l,ωˆ ), (15.291) ∝ −∞ 2π sinhω ¯ Z with ωˆ2 ωˆ4 ωˆ6 V (l, ωˆ2) V (l)+ V¯ (l, ωˆ2)= V (l)+ V (l) + V (l) + V (l) + .... (15.292) ≡ 0 0 1 l 2 l2 3 l3 The ﬁrst term

2 d 1 d 9 2 (d 1) 32 13 d +5 d 3 V0(l) = − l + − l + − − l − 6 360 6480 34 272 d + 259 d2 110 d3 + 25 d4 − − l4 + ... (15.293) − 259200 contributes only to the normalization of P (r; l), and can be omitted in (15.291). The remainder has the expansion coeﬃcients

2 3 d 3 2 ( 5+9 d) 3 455 + 431 d + 91 d +5 d 4 V1(l) = − l + − l + − l + ..., − 360 7560 ( 1+ d) 907200 (d 1)2 − − (5 3 d) l3 31+42 d + 25 d2 l4 V (l) = − − + ..., (15.294) 2 − 7560 − 907200 (d 1) − (d 1) l4 V (l) = − + .... 3 − 18900 In the physical most interesting case of three dimensions, the ﬁrst nonzero correction arises to order l3. This explains the remarkable agreement of the moments in (15.234) and (15.234) up to order l2. The correction terms V¯ (l, ωˆ2) may be included perturbatively into the sum over k in Eq. (15.220) by noting that the expectation value of powers ofω ˆ2/l within theω ˆ-integral (15.221) are

2k + (D 1)/2 2 3 ωˆ2/l = a2 − , ωˆ2/l =3a4, ωˆ2/l = 15a6, (15.295) k ≡ (D 1)(1 r) k k − − so that we obtain an extra factor e−fk 4 f = V (l)a2 + 3V (l) V 2(l) a4 + 15V (l) 12V (l)V (l)+ V 3(l) a6 + ..., (15.296) k 1 k 2 − 1 k 3 − 1 2 3 1 k where up to order l4:

3D 5 156 231D 26D2 7D3 3V (l) V 2(l) = − l3 + − − − l4 + ..., 2 − 1 2520 907200 (D 1) − 4 D 1 15V (l) 12V (l)V (l)+ V 3(l) = − l4 + .... (15.297) 3 − 1 2 3 1 − 1260 1074 15 Path Integrals in Polymer Physics

15.10 Excluded-Volume Eﬀects

A significant modification of these properties is brought about by the interactions between the chain elements. If two of them come close to each other, the molecular forces prevent them from occupying the same place. This is called the excluded- volume effect. In less than four dimensions, it gives rise to a scaling law for the expectation value R2 as a function of L: h i R2 L2ν , (15.298) h i ∝ as stated in (15.38). The critical exponent ν is a number between the random-chain value ν =1/2 and the stiff-chain value ν = 1. To derive this behavior we consider the polymer in the limiting path integral approximation (15.80) to a random chain which was derived for R2/La 1 and which is very accurate whenever the probability distribution is sizable.≪ Thus we start with the time-sliced expression

1 N−1 dDx P (R)= n exp N /h¯ , (15.299) N D  D  −A 2πa/M n=1 Z 2πa/M Y   q  q  with the action N M (∆x )2 N = a n , (15.300) A 2 a2 nX=1 and the mass parameter (15.79). In the sequel we use natural units in which energies are measured in units of kBT , and write down all expressions in the continuum limit. The probability (15.299) is then written as

D −AL[x] PL(R)= x e , (15.301) Z D where we have used the label L = Na rather than N. From the discussion in the previous section we know that although this path integral represents an ideal random chain, we can also account for a finite stiffness by interpreting the number a as an effective length parameter aeff given by (15.158). The total Euclidean time in the path integral τ τ =¯h/k T corresponds to the total length of the polymer b − a B L. We now assume that the molecules of the polymer repel each other with a two- body potential V (x, x′). Then the action in the path integral (15.301) has to be supplemented by an interaction

L L 1 ′ ′ int = dτ dτ V (x(τ), x(τ )). (15.302) A 2 Z0 Z0 Note that the interaction is of a purely spatial nature and does not depend on the parameters τ, τ ′, i.e., it does not matter which two molecules in the chain come close to each other.

H. Kleinert, PATH INTEGRALS 15.10 Excluded-Volume Eﬀects 1075

The effects of an interaction of this type are most elegantly calculated by making use of a Hubbard-Stratonovich transformation. Generalizing the procedure in Subsection 7.15.1, we introduce an auxiliary fluctuating field variable ϕ(x) at every space point x and replace by Aint L ϕ 1 D D ′ −1 ′ ′ int = dτ ϕ(x(τ)) d xd x ϕ(x)V (x, x )ϕ(x ). (15.303) A Z0 − 2 Z Here V −1(x, x′) denotes the inverse of V (x, x′) under functional multiplication, defined by the integral equation

dDx′ V −1(x, x′)V (x′, x′′)= δ(D)(x x′′). (15.304) Z − To see the equivalence of the action (15.303) with (15.302), we rewrite (15.303) as

ϕ D 1 D D ′ −1 ′ ′ int = d x ρ(x)ϕ(x) d xd x ϕ(x)V (x, x )ϕ(x ), (15.305) A Z − 2 Z where ρ(x) is the particle density

L ρ(x) dτ δ(D)(x x(τ)). (15.306) ≡ Z0 − Then we perform a quadratic completion to 1 ϕ = dDxdDx′ ϕ′(x)V −1(x, x′)ϕ′(x′) ρ(x)V (x, x′)ρ(x′) , (15.307) Aint −2 − Z h i with the shifted ﬁeld

ϕ′(x) ϕ(x) dDx′ V (x, x′)ρ(x′). (15.308) ≡ − Z Now we perform the functional integral

−Aϕ ϕ(x) e int (15.309) Z D integrating ϕ(x) at each point x from i to i along the imaginary ﬁeld axis. − ∞ ∞ The result is a constant functional determinant [det V −1(x, x′)]−1/2. This can be ignored since we shall ultimately normalize the end-to-end distribution to unity. Inserting (15.306) into the surviving second term in (15.307), we obtain precisely the original interaction (15.302). Thus we may study the excluded-volume problem by means of the equivalent path integral

D −A PL(R) x(τ) ϕ(x) e , (15.310) ∝ Z D Z D where the action is given by the sum A = L[x, x˙ ,ϕ]+ [ϕ], (15.311) A A A 1076 15 Path Integrals in Polymer Physics of the line and field actions L M L[x,ϕ] dτ x˙ 2 + ϕ(x(τ)) , (15.312) A ≡ Z0 2 1 [ϕ] dDxdDx′ ϕ(x)V −1(x, x′)ϕ(x′), (15.313) A ≡ −2 Z respectively. The path integral (15.310) over x(τ) and ϕ(x) has the following physical interpretation. The line action (15.312) describes the orbit of a particle in a space-dependent random potential ϕ(x). The path integral over x(τ) yields the end- to-end distribution of the fluctuating polymer in this potential. The path integral over all potentials ϕ(x) with the weight e−A[ϕ] accounts for the repulsive cloud of the fluctuating chain elements. To be convergent, all ϕ(x) integrations in (15.310) have to run along the imaginary field axis. To evaluate the path integrals (15.310), it is useful to separate x(τ)- and ϕ(x)- integrations and to write end-to-end distributions as an average over ϕ-fluctuations

−A[ϕ] ϕ PL(R) ϕ(x) e PL (R, 0), (15.314) ∝ Z D where

ϕ D −AL[x,ϕ] PL (R, 0)= x(τ) e (15.315) Z D is the end-to-end distribution of a random chain moving in a fixed external potential ϕ ϕ(x). The presence of this potential destroys the translational invariance of PL . This is why we have recorded the initial and final points 0 and R. In the final distribution PL(R) of (15.314), the invariance is of course restored by the integration over all ϕ(x). ϕ It is possible to express the distribution PL (R, 0) in terms of solutions of an associated Schrödinger equation. With the action (15.312), this equation is obviously

∂ 1 2 ϕ (D) ∂R + ϕ(R) PL (R, 0)= δ (R 0)δ(L). (15.316) "∂L − 2M # −

ϕ If ψE(R) denotes the time-independent solutions of the Hamiltonian operator

ϕ 1 2 Hˆ = ∂R + ϕ(R), (15.317) −2M ϕ the probability PL (R) has a spectral representation of the form

ϕ −EL ϕ ϕ ∗ PL (R, 0)= dEe ψE(R)ψE (0), L> 0. (15.318) Z From now on, we assume the interaction to be dominated by the simplest possible repulsive potential proportional to a δ-function:

V (x, x′)= vaDδ(D)(x x′). (15.319) −

H. Kleinert, PATH INTEGRALS 15.10 Excluded-Volume Eﬀects 1077

Then the functional inverse is

V −1(x, x′)= v−1a−Dδ(D)(x x′), (15.320) − and the ϕ-action (15.312) reduces to

v−1a−D [ϕ]= dDxϕ2(x). (15.321) A − 2 Z The path integrals (15.314), (15.315) can be solved approximately by applying the semiclassical methods of Chapter 4 to both the x(τ)- and the ϕ(x)-path integrals. These are dominated by the extrema of the action and evaluated via the leading saddle point approximation. In the ϕ(x)-integral, the saddle point is given by the equation δ v−1a−Dϕ(x)= log P ϕ(R, 0). (15.322) δϕ(x) L This is the semiclassical approximation to the exact equation

L v−1a−D ϕ(x) = ρ(x) dτ δ(D)(x x(τ)) , (15.323) h i h i ≡ *Z0 − +x where ... x is the average over all line ﬂuctuations calculated with the help of the probabilityh i distribution (15.315). The exact equation (15.323) follows from a functional diﬀerentiation of the path ϕ integral for PL with respect to ϕ(x):

δ ϕ δ D −AL[x,ϕ]−A[ϕ] PL (R)= ϕ x e =0. (15.324) δϕ(x) Z D δϕ(x) Z D By anchoring one end of the polymer at the origin and carrying the path integral from there to x(τ), and further on to R, the right-hand side of (15.323) can be expressed as a convolution integral over two end-to-end distributions:

L L (D) ′ ϕ ϕ dτ δ (x x(τ)) = dL PL′ (x)PL−L′ (R x). (15.325) *Z0 − +x Z0 − With (15.323), this becomes

L −1 −D ′ ϕ ϕ v a ϕ(x) x = dL PL′ (x)PL−L′ (R x), (15.326) h i Z0 − which is the same as (15.323). According to Eq. (15.322), the extremal ϕ(x) depends really on two variables, x and R. This makes the solution diﬃcult, even at the semiclassical level. It becomes simple only for R = 0, i.e., for a closed polymer. Then only the variable x remains and, by rotational symmetry, ϕ(x) can depend only on r = x . For R = 0, on the | | 6 1078 15 Path Integrals in Polymer Physics other hand, the rotational symmetry is distorted to an ellipsoidal geometry, in which a closed-form solution of the problem is hard to ﬁnd. As an approximation, we may use a rotationally symmetric ansatz ϕ(x) ϕ(r) also for R = 0 and calculate the ≈ 6 end-to-end probability distribution PL(R) via the semiclassical approximation to the two path integrals in Eq. (15.310). The saddle point in the path integral over ϕ(x) gives the formula [compare (15.314)]

L ϕ D M 2 PL(R) PL (R, 0)= x exp dτ x˙ + ϕ(r(τ)) . (15.327) ∼ Z D (−Z0 2 ) Thereby it is hoped that for moderate R, the error is small enough to justify this approximation. Anyhow, the analytic results supply a convenient starting point for better approximations. Neglecting the ellipsoidal distortion, it is easy to calculate the path integral over ϕ x(τ) for PL (R, 0) in the saddle point approximation. At an arbitrary given ϕ(r), we must find the classical orbits. The Euler-Lagrange equation has the first integral of motion M x˙ 2 ϕ(r)= E = const. (15.328) 2 − At fixed L, we have to find the classical solutions for all energies E and all angular momenta l. The path integral reduces an ordinary double integral over E and l which, in turn, is evaluated in the saddle point approximation. In a rotationally symmetric potential ϕ(r), the leading saddle point has the angular momentum l =0 corresponding to a symmetric polymer distribution. Then Eq. (15.328) turns into a purely radial differential equation dr dτ = . (15.329) 2[E + ϕ(r)]/M q For a polymer running from the origin to R we calculate R dr L = . (15.330) Z0 2[E + ϕ(r)]/M q This determines the energy E as a function of L. It is a functional of the yet unknown field ϕ(r):

E = EL[ϕ]. (15.331) The classical action for such an orbit can be expressed in the form L M 2 cl[x,ϕ] = dτ x˙ + ϕ(r(τ)) A Z0 2 L M L = dτ x˙ 2 ϕ(r(τ)) + dτMx˙ 2 − Z0 2 − Z0 R = EL + dr 2M[E + ϕ(r)]. (15.332) − 0 Z q

H. Kleinert, PATH INTEGRALS 15.10 Excluded-Volume Eﬀects 1079

In this expression, we may consider E as an independent variational parameter. The relation (15.330) between E,L,R,ϕ(r), by which E is ﬁxed, reemerges when extremizing the classical expression [x,ϕ]: Acl ∂ [x, x˙ ,ϕ]=0. (15.333) ∂E Acl The classical approximation to the entire action [x,ϕ]+ [ϕ] is then A A r ′ ′ 1 −1 −D D 2 cl = EL + dr 2M[E + ϕ(r )] v a d xϕ (r). (15.334) A − 0 − 2 Z q Z This action is now extremized independently in ϕ(r), E. The extremum in ϕ(r) is obviously given by the algebraic equation

′ ′ 0 r > r, ϕ(r )= D −1 ′ 1−D ′ for ′ (15.335) ( Mva SD r / 2M[E + ϕ(r )] r < r, q which is easily solved. We rewrite it as

E + ϕ(r)= ξ3ϕ−2(r), (15.336) with the abbreviation

ξ3 = αr−2δ, (15.337) where

δ D 1 > 0 (15.338) ≡ − and M α v2a2DS−2. (15.339) ≡ 2 D For large ξ 1/E, i.e., small r α2/δE−6/δ, we expand the solution as follows ≫ ≪ E E2 ϕ(r)= ξ + + .... (15.340) − 3 9 This expansion is reinserted into the classical action (15.334), making it a power series in E. A further extremization in E yields E = E(L, r). The extremal value of the action yields an approximate distribution function of a monomer in the closed polymer (which runs through the origin):

P (R) e−Acl(L,R). (15.341) L ∝ To see how this happens consider ﬁrst the noninteracting limit where v = 0. Then the solution of (15.335) is ϕ(r) 0, and the classical action (15.334) becomes ≡ = EL + √2MER. (15.342) Acl − 1080 15 Path Integrals in Polymer Physics

The extremization in E gives

M R2 E = , (15.343) 2 L2 yielding the extremal action

M R2 = . (15.344) Acl 2 L The approximate distribution is therefore

2 P (R) e−Acl = e−MR /2L. (15.345) N ∝ The interacting case is now treated in the same way. Using (15.336), the classical action (15.334) can be written as

R M ′ 1 3/2 1 cl = EL + dr ϕ + E ξ . (15.346) A − s 2 0 " − 2 ϕ + E # Z q By expanding this action in a power series in E [after having inserted (15.340) for ϕ], we obtain with ǫ(r) E/ξ = Eα−1/3r2δ/3 ≡ R M 1/6 ′ ′−δ/3 3 ′ 1 2 ′ cl = EL + α dr r + ǫ(r ) ǫ (r )+ ... . (15.347) s A − 2 Z0 2 − 6 As long as δ < 3, i.e., for

D< 4, (15.348) the integral exists and yields an expansion 1 = EL + a (R)+ a (R)E a (R)E2 + ..., (15.349) Acl − 0 1 − 2 2 with the coeﬃcients

M 9 1−δ/3 1/6 1 a0(R) = R α , − 2 s2 δ 3 − M 1+δ/3 −1/6 1 a1(R) = 3 R α , s 2 δ +3

1 M 1+δ −1/2 1 a2(R) = R α . (15.350) 3s 2 δ +1 Extremizing in E gives the action Acl

1 2 cl = a0(R)+ [L a1(R)] + .... (15.351) A 2a2(R) −

H. Kleinert, PATH INTEGRALS 15.11 Flory’s Argument 1081

The approximate end-to-end distribution function is therefore

1 2 PL(R) exp a0(R) [L a1(R)] , (15.352) ≈N (− − 2a2(R) − ) where is an appropriate normalization factor. The distribution is peaked around N M 1 L =3 R1+δ/3α−1/6 . (15.353) s 2 δ +3 This shows the most important consequence of the excluded-volume eﬀect: The average value of R2 grows like

6/(D+2) D +2 2 R2 α1/(D+2) L . (15.354) h i ≈  3 sM    Thus we have found a scaling law of the form (15.298) with the critical exponent 3 ν = . (15.355) D +2 The repulsion between the chain elements makes the excluded-volume chain reach out further into space than a random chain [although less than a completely stiﬀ chain, which is always reached by the solution (15.354) for D = 1]. The restriction D < 4 in (15.348) is quite important. The value Duc = 4 (15.356) is called the upper critical dimension. Above it, the set of all possible intersections of a random chain has the measure zero and any short-range repulsion becomes irrelevant. In fact, for D > Duc it is possible to show that the polymer behaves like a random chain without any excluded-volume eﬀect satisfying R2 L. h i ∝ 15.11 Flory’s Argument

Once we expect a power-like scaling law of the form R2 L2ν , (15.357) h i ∝ the critical exponent (15.355) can be derived from a very simple dimensional argument due to Flory. We take the action L M vaD L L = dτ x˙ 2 dτ dτ ′ δ(D)(x(τ) x(τ ′)), (15.358) A Z0 2 − 2 Z0 Z0 − with M = D/a, and replace the two terms by their dimensional content, L for the τ-variable and R- for each x-component. Then M R2 vaD L2 L . (15.359) A ∼ 2 L2 − 2 RD 1082 15 Path Integrals in Polymer Physics

Extremizing this expression in R at ﬁxed L gives

R R−D−1L2, (15.360) L ∼ implying

R2 L6/(D+2), (15.361) ∼ and thus the critical exponent (15.355).

15.12 Polymer Field Theory

There exists an alternative approach to finding the power laws caused by the excluded-volume effects in polymers which is superior to the previous one. It is based on an intimate relationship of polymer fluctuations with field fluctuations in a certain somewhat artificial and unphysical limit. This limit happens to be accessible to approximate analytic methods developed in recent years in quantum field theory. According to Chapter 7, the statistical mechanics of a many-particle ensemble can be described by a single fluctuating field. Each particle in such an ensemble moves through spacetime along a fluctuating orbit in the same way as a random chain does in the approximation (15.80) to a polymer in Section 15.6. Thus we can immediately conclude that ensembles of polymers may also be described by a single fluctuating field. But how about a single polymer? Is it possible to project out a single polymer of the ensemble in the field-theoretic description? The answer is positive. We start with the result of the last section. The end-to-end distribution of the polymer in the excluded-volume problem is rewritten as an integral over the fluctuating field ϕ(x):

−A[ϕ] ϕ PL(xb, xa)= ϕ e PL (xb, xa), (15.362) Z D with an action for the auxiliary ϕ(x) ﬁeld [see (15.312)]

1 [ϕ]= dDxdDx′ϕ(x)V −1(x, x′)ϕ(x′), (15.363) A −2 Z and an end-to-end distribution [see (15.315)]

L ϕ M 2 PL (xb, xa)= x exp dτ x˙ + ϕ(x(τ)) , (15.364) Z D (− Z0 2 ) which satisﬁes the Schr¨odinger equation [see (15.316)]

∂ 1 ∇2 ϕ ′ (3) ′ + ϕ(x) PL (x, x )= δ (x x )δ(L). (15.365) "∂L − 2M # −

H. Kleinert, PATH INTEGRALS 15.12 Polymer Field Theory 1083

ϕ Since PL and PL vanish for L < 0, it is convenient to go over to the Laplace transforms ∞ ′ 1 −Lm2/2M ′ Pm2 (x, x ) = dLe PL(x, x ), (15.366) 2M Z0 ∞ ϕ ′ 1 −Lm2/2M ϕ ′ Pm2 (x, x ) = dLe PL (x, x ). (15.367) 2M Z0 The latter satisﬁes the L-independent equation

2 2 ϕ ′ (3) ′ [ ∇ + m +2Mϕ(x)]P 2 (x, x )= δ (x x ). (15.368) − m − The quantity m2/2M is, of course, just the negative energy variable E in (15.332): − m2 E . (15.369) − ≡ 2M ϕ ′ The distributions Pm2 (x, x ) describe the probability of a polymer of any length running from x′ to x, with a Boltzmann-like factor e−Lm2/2M governing the distribution of lengths. Thus m2/2M plays the role of a chemical potential. We now observe that the solution of Eq. (15.368) can be considered as the correlation function of an auxiliary ﬂuctuating complex ﬁeld ψ(x):

ϕ ′ ϕ ′ ∗ ′ P 2 (x, x ) = G (x, x )= ψ (x)ψ(x ) m 0 h iϕ ψ∗(x) ψ(x) ψ∗(x)ψ(x′) exp [ψ∗,ψ,ϕ] D D {−A }, (15.370) ≡ ψ∗(x) ψ(x) exp [ψ∗,ψ,ϕ] R D D {−A } with a field action R [ψ∗,ψ,ϕ]= dDx ∇ψ∗(x)∇ψ(x)+ m2ψ∗(x)ψ(x)+2Mϕ(x)ψ∗(x)ψ(x) .(15.371) A Z n o The second part of Eq. (15.370) defines the expectations ... . In this way, we h iψ express the Laplace-transformed distribution Pm2 (xb, xa) in (15.366) in the purely field-theoretic form

′ ∗ ′ Pm2 (x, x )= ϕ exp [ϕ] ψ (x)ψ(x ) ϕ Z D {−A }h i 1 = ϕ exp dDydDy′ϕ(y)V −1(y, y′)ϕ(y′) Z D 2 Z ψ∗ ψ ψ∗(x)ψ(x′) exp [ψ∗,ψ,ϕ] D D {−A }. (15.372) × ψ∗ ψ exp [ψ∗,ψ,ϕ] R D D {−A } It involves only a fluctuatingR fieldR which contains all information on the path fluctuations. The field ψ(x) is, of course, the analog of the second-quantized field in Chapter 7. Consider now the probability distribution of a single monomer in a closed polymer chain. Inserting the polymer density function

L ρ(R) dτδ(D)(R x(τ)) (15.373) ≡ Z0 − 1084 15 Path Integrals in Polymer Physics into the original path integral for a closed polymer

L D (D) PL(R)= dτ x ϕ exp L [ϕ] δ (R x(τ)), (15.374) Z0 Z D Z D {−A −A } − the δ-function splits the path integral into two parts

L ϕ ϕ PL(R)= ϕ exp [ϕ] dτPL−τ (0, R)Pτ (R, 0). (15.375) Z D {−A } Z0 When going to the Laplace transform, the convolution integral factorizes, yielding

ϕ ϕ Pm2 (R)= ϕ(x) exp [ϕ] Pm2 (0, R)Pm2 (R, 0). (15.376) Z D {−A }

With the help of the ﬁeld-theoretic expression for Pm2 (R) in Eq. (15.370), the product of the correlation functions can be rewritten as

ϕ ϕ ∗ ∗ P 2 (0, R)P 2 (0, R)= ψ (R)ψ(0) ψ (0)ψ(R) . (15.377) m m h iϕh iϕ We now observe that the ﬁeld ψ appears only quadratically in the action [ψ∗,ψ,ϕ]. The product of correlation functions in (15.377) can therefore be viewedA as a term in the Wick expansion (recall Section 3.10) of the four-ﬁeld correlation function

ψ∗(R)ψ(R)ψ∗(0)ψ(0) . (15.378) h iϕ This would be equal to the sum of pair contractions

ψ∗(R)ψ(R) ψ∗(0)ψ(0) + ψ∗(R)ψ(0) ψ∗(0)ψ(R) . (15.379) h iϕh iϕ h iϕh iϕ There are no contributions containing expectations of two ψ or two ψ∗ fields which could, in general, appear in this expansion. This allows the right-hand side of (15.377) to be expressed as a difference between (15.378) and the first term of (15.379):

ϕ ϕ ∗ ∗ ∗ ∗ P 2 (0, R)P 2 (0, R)= ψ (R)ψ(R)ψ (0)ψ(0) ψ (R)ψ(R) ψ (0)ψ(0) . m m h iϕ −h iϕh iϕ (15.380)

The right-hand side only contains correlation functions of a collective ﬁeld, the density ﬁeld [8]

ρ(R)= ψ∗(R)ψ(R), (15.381) in terms of which

ϕ ϕ P 2 (0, R)P 2 (0, R)= ρ(R)ρ(0) ρ(R) ρ(0) . (15.382) m m h iϕ −h iϕh iϕ Now, the right-hand side is the connected correlation function of the density ﬁeld ρ(R):

ρ(R)ρ(0) ρ(R)ρ(0) ρ(R) ρ(0) . (15.383) h iϕ,c ≡h iϕ −h iϕh iϕ

H. Kleinert, PATH INTEGRALS 15.12 Polymer Field Theory 1085

In Section 3.10 we have shown how to generate all connected correlation functions: The action [ψ, ψ∗,ϕ] is extended by a source term in the density ﬁeld ρ(x) A

∗ D D ∗ source[ψ ,ψ,K]= d xK(x)ρ(x)= d xK(x)ψ (x)ψ(x)), (15.384) A − Z Z and one considers the partition function

∗ ∗ ∗ Z[K,ϕ] ψ ψ exp [ψ ,ψ,ϕ] source[ψ ,ψ,K] . (15.385) ≡ Z D D {−A −A } This is the generating functional of all correlation functions of the density ﬁeld ρ(R)= ψ∗(R)ψ(R) at a ﬁxed ϕ(x). They are obtained from the functional derivatives

−1 δ δ ρ(x1) ρ(xn) ϕ = Z[K,ϕ] Z[K,ϕ] . h ··· i δK(x ) ··· δK(x ) K=0 1 n

(15.386)

Recalling Eq. (3.559), the connected correlation functions of ρ(x) are obtained sim- ilarly from the logarithm of Z[K,ϕ]:

δ δ ρ(x1) ρ(xn) ϕ,c = log Z[K,ϕ] . h ··· i δK(x ) ··· δK(x ) K=0 1 n

(15.387)

For n = 2, the connectedness is seen directly by performing the diﬀerentiations according to the chain rule:

δ δ ρ(R)ρ(0) ϕ,c = log Z[K,ϕ] h i δK(R) δK(0) K=0

δ δ = Z−1[K,ϕ] Z [K,ϕ] δK(R) δK(0) K=0

= ρ(R)ρ(0) ρ(R) ρ(0) . (15.388) h iϕ −h iϕh iϕ This agrees indeed with (15.383). We can therefore rewrite the product of Laplace- transformed distributions (15.382) at a ﬁxed ϕ(x) as

ϕ ϕ δ δ P 2 (0, R)P 2 (0, R)= log Z[K,ϕ] . (15.389) m m δK(R) δK(0) K=0

The Laplace-transformed monomer distribution (15.376) is then obtained by averaging over ϕ(x), i.e., by the path integral

δ δ Pm2 (R)= ϕ(x) exp [ϕ] log Z[K,ϕ] . (15.390) δK(R) δK(0) D {−A } K=0 Z

1086 15 Path Integrals in Polymer Physics

Were it not for the logarithm in front of Z, this would be a standard calculation of correlation functions within the combined ψ,ϕ ﬁeld theory whose action is

[ψ∗,ψ,ϕ] = [ψ∗,ψ,ϕ]+ [ϕ] Atot A A = dDx ∇ψ∗∇ψ + m2ψ∗ψ +2Mϕψ∗ψ Z 1 dDxdDx′ϕ(x)V −1(x, x′)ϕ(x′). (15.391) − 2 Z To account for the logarithm we introduce a simple mathematical device called the replica trick [9]. We consider log Z[K,ϕ] in (15.388)–(15.390) as the limit

1 log Z = lim (Zn 1) , (15.392) n→0 n − and observe that the nth power of the generating functional, Zn, can be thought of as arising from a field theory in which every field ψ occurs n times, i.e., with n identical replica. Thus we add an extra internal symmetry label α =1,...,n to the fields ψ(x) and calculate Zn formally as

n ∗ ∗ ∗ Z [K,ϕ]= ψα ψα exp [ψα, ψα,ϕ] [ϕ] source[ψα, ψα,K] , Z D D {−A −A −A } (15.393) with the replica ﬁeld action

[ψ , ψ∗ ,ϕ]= dDx ∇ψ∗ ∇ψ + m2ψ∗ ψ +2Mϕ ψ∗ ψ , (15.394) A α α α α α α α α Z and the source term

∗ D ∗ source[ψα, ψα,K]= d xψα(x)ψα(x)K(x). (15.395) A − Z A sum is implied over repeated indices α. By construction, the action is symmetric under the group U(n) of all unitary transformations of the replica ﬁelds ψα. In the partition function (15.393), it is now easy to integrate out the ϕ(x)- ﬂuctuations. This gives

n ∗ n ∗ ∗ Z [K,ϕ]= ψα ψα exp [ψα, ψα] source[ψα, ψα,K] , (15.396) Z D D {−A −A } with the action

n[ψ∗ , ψ ] = dDx ∇ψ∗ ∇ψ + m2ψ∗ ψ A α α α α α α Z 1 2 D D ′ ∗ ′ ∗ ′ ′ + (2M) d xd x ψα (x)ψα(x)V (x, x )ψβ (x )ψβ(x ). (15.397) 2 Z It describes a self-interacting ﬁeld theory with an additional U(n) symmetry.

H. Kleinert, PATH INTEGRALS 15.12 Polymer Field Theory 1087

In the special case of a local repulsive potential V (x, x′) of Eq. (15.319), the second term becomes simply

∗ 1 2 D D ∗ 2 int[ψα, ψα]= (2M) va d x [ψα (x)ψα(x)] . (15.398) A 2 Z Using this action, we can ﬁnd log Z[K,ϕ] via (15.392) from the functional integral

1 ∗ n ∗ ∗ log Z[K,ϕ] lim ψ ψα exp [ψ , ψα] source[ψ , ψα,K] 1 . ≡ n→0 n D αD {−A α −A α } − Z (15.399) This is the generating functional of the Laplace-transformed distribution (15.390) which we wanted to calculate. A polymer can run along the same line in two orientations. In the above description with complex replica fields it was assumed that the two orientations can be distinguished. If they are indistinguishable, the polymer fields Ψα(x) have to be taken as real. Such a field-theoretic description of a fluctuating polymer has an important advantage over the initial path integral formulation based on the analogy with a particle orbit. It allows us to establish contact with the well-developed theory of critical phenomena in field theory. The end-to-end distribution of long polymers at large L is determined by the small-E regime in Eqs. (15.330)–(15.349), which corresponds to the small-m2 limit of the system here [see (15.369)]. This is precisely the regime studied in the quantum field-theoretic approach to critical phenomena in many-body systems [10, 11]. It can be shown that for D larger than the upper critical dimension Duc = 4, the behavior for m2 0 of all Green functions coincides with the free-field behavior. For D = Duc, this behavior→ can be deduced from scale invariance arguments of the action, using naive dimensional counting arguments. The fluctuations turn out to cause only logarithmic corrections to the scale-invariant power laws. One of the main developments in quantum field theory in recent years was the discovery that the scaling powers for D < Duc can be calculated via an expansion of all quantities in powers of

ǫ = Duc D, (15.400) − the so-called ǫ-expansion. The ǫ-expansion for the critical exponent ν which rules the relation between R2 and the length of a polymer L, R2 L2ν , can be derived from a real φ4-theory with n replica as follows [12]: h i ∝

ν−1 = 2+ (n+2)ǫ 1 ǫ (13n + 44) n+8 − − 2(n+8)2 2 + ǫ [3n3 n452n2 2672n 5312 8(n+8)4 − − − + ζ(3)(n + 8) 96(5n + 22)] 3 · + ǫ [3n5 + 398n4 12900n3 81552n2 219968n 357120 32(n+8)6 − − − − + ζ(3)(n + 8) 16(3n4 194n3 + 148n2 + 9472n + 19488) · − + ζ(4)(n + 8)3 288(5n + 22) · 1088 15 Path Integrals in Polymer Physics

ζ(5)(n + 8)2 1280(2n2 + 55n + 186)] 4 − · + ǫ [3n7 1198n6 27484n5 1055344n4 128(n+8)8 − − − 5242112n3 5256704n2 + 6999040n 626688 − − − ζ(3)(n + 8) 16(13n6 310n5 + 19004n4 + 102400n3 − · − 381536n2 2792576n 4240640) − − − ζ2(3)(n + 8)2 1024(2n4 + 18n3 + 981n2 + 6994n + 11688) − · + ζ(4)(n + 8)3 48(3n4 194n3 + 148n2 + 9472n + 19488) · − + ζ(5)(n + 8)2 256(155n4 + 3026n3 + 989n2 66018n 130608) · − − ζ(6)(n + 8)4 6400(2n2 + 55n + 186) − · + ζ(7)(n + 8)3 56448(14n2 + 189n + 526)] , (15.401) · o where ζ(x) is Riemann’s zeta function (2.521). As shown above, the single-polymer properties must emerge in the limit n 0. There, ν−1 has the ǫ-expansion → ǫ 11 ν−1 =2 ǫ2 +0.114 425 ǫ3 0.287 512 ǫ4 +0.956 133 ǫ5. (15.402) − 4 − 128 − This is to be compared with the much simpler result of the last section D +2 ǫ ν−1 = =2 . (15.403) 3 − 3 A term-by-term comparison is meaningless since the field-theoretic ǫ-expansion has a grave problem: The coefficients of the ǫn-terms grow, for large n, like n!, so that the series does not converge anywhere! Fortunately, the signs are alternating and the series can be resummed [13]. A simple first approximation used in ǫ-expansions is to re-express the series (15.402) as a ratio of two polynomials of roughly equal degree 2. +1.023 606 ǫ 0.225 661 ǫ2 ν−1 = − , (15.404) |rat 1. +0.636 803 ǫ 0.011 746 ǫ2 +0.002 677 ǫ3 − called a Padéapproximation. Its Taylor coefficients up to ǫ5 coincide with those of the initial series (15.402). It can be shown that this approximation would converge with increasing orders in ǫ towards the exact function represented by the divergent series. In Fig. 15.7, we plot the three functions (15.403), (15.402), and (15.404), the last one giving the most reliable approximation ν−1 0.585. (15.405) ≈ Note that the simple Flory curve lies very close to the Padécurve whose calculation requires a great amount of work. There exists a general scaling relation between the exponent ν and another exponent appearing in the total number of polymer configurations of length L which behaves like S = Lα−2. (15.406)

H. Kleinert, PATH INTEGRALS 15.13 Fermi Fields for Self-Avoiding Lines 1089

Figure 15.7 Comparison of critical exponent ν in Flory approximation (dashed line) with result of divergent ǫ-expansion obtained from quantum ﬁeld theory (dotted line) and its Pad´eresummation (solid line). The value of the latter gives the best approximation ν 0.585 at ǫ = 1. ≈

The relation is

α =2 Dν. (15.407) − Direct enumeration studies of random chains on a computer suggest a number 1 α , (15.408) ∼ 4 corresponding to ν =7/12 0.583, very close to (15.405). The Flory estimate for the≈ exponent α reads, incidentally, 4 D α = − . (15.409) D +2 In three dimensions, this yields α =1/5, not too far from (15.408). The discrepancies arise from inaccuracies in both treatments. In the ﬁrst treatment, they are due to the use of the saddle point approximation and the fact that the δ-function does not completely rule out the crossing of the lines, as required by the true self-avoidance of the polymer. The ﬁeld theoretic ǫ-expansion, on the other hand, which in principle can give arbitrarily accurate results, has the problem of being divergent for any ǫ. Resummation procedures are needed and the order of the expansion must be quite large ( ǫ5) to extract reliable numbers. ≈ 15.13 Fermi Fields for Self-Avoiding Lines

There exists another way of enforcing the self-avoiding property of random lines [14]. It is based on the observation that for a polymer field theory with n fluctuating complex fields Ψα and a U(n)-symmetric fourth-order self-interaction as in the action 1090 15 Path Integrals in Polymer Physics

(15.397), the symmetric incorporation of a set of m anticommuting Grassmann fields removes the effect of m of the Bose fields. For free fields this observation is trivial since the functional determinant of Bose and Fermi fields are inverse to one-another. In the presence of a fourth-order self-interaction, where the replica action has the form (15.397), we can always go back, by a Hubbard-Stratonovich transformation, to the action involving the auxiliary field ϕ(x) in the exponent of (15.393). This exponent is purely quadratic in the replica field, and each path integral over a Fermi field cancels a functional determinant coming from the Bose field. This boson-destructive effect of fermions allows us to study theories with a negative number of replica. We simply have to use more Fermi than Bose fields. More- over, we may conclude that a theory with n = 2 has necessarily trivial critical exponents. From the above arguments it is equivalent− to a single complex Fermi field theory with fourth-order self-interaction. However, for anticommuting Grass- mann fields, such an interaction vanishes: (θ†θ)2 = [(θ iθ )(θ + iθ )]2 = [2iθ θ ]2 =0. (15.410) 1 − 2 1 2 1 2 Looking back at the ǫ-expansion for the critical exponent ν in Eq. (15.401) we can verify that up to the power ǫ5 all powers in ǫ do indeed vanish and ν takes the mean-field value 1/2.

Appendix 15A Basic Integrals

∆(0, 0) = ∆(L,L)= L/3, (15A.1) L 2 I1 = ds ∆(s,s)= L /6, (15A.2) Z0 L 2 I2 = ds ˙∆ (s,s)= L/12, (15A.3) Z0 L 2 3 I3 = ds ∆ (s,s)= L /30, (15A.4) Z0 L 2 2 I4 = ds ∆(s,s) ˙∆ (s,s)=7L /360, (15A.5) Z0 L L ′ 2 ′ 4 I5 = ds ds ∆ (s,s )= L /90, (15A.6) Z0 Z0 ∆(0,L) = ∆(L, 0) = L/6, (15A.7) L − 2 2 3 I6 = ds ∆ (s, 0)+∆ (s,L) =2L /45, (15A.8) Z0 L L ′ 2 ′ 3 I7 = ds ds ∆(s,s)˙∆ (s,s )= L /45, (15A.9) Z0 Z0 L L ′ ′ ′ 3 I8 = ds ds ˙∆(s,s)∆(s,s )˙∆(s,s )= L /180, (15A.10) Z0 Z0 L 2 2 2 I9 = ds ∆(s,s) ˙∆ (s, 0) + ˙∆ (s,L) = 11L /90, (15A.11) Z0 L 2 I10 = ds ˙∆(s,s) [∆(s, 0)˙∆(s, 0) + ∆(s,L)˙∆(s,L)] = 17L /360, (15A.12) Z0

H. Kleinert, PATH INTEGRALS Appendix 15B Loop Integrals 1091

˙∆(0, 0) = ˙∆(L,L)= 1/2, (15A.13) − L L − ′ ′ ′ ′ 3 I11 = ds ds ∆(s,s) ∆˙(s,s )˙∆(s ,s )= L /360, (15A.14) Z0 Z0 L L ′ ′ 2 ′ 3 I12 = ds ds ∆(s,s )˙∆ (s,s )= L /90. (15A.15) Z0 Z0 Appendix 15B Loop Integrals

We list here the Feynman integrals evaluated with dimensional regularization rules whenever necessary. Depending whether they occur in the calculation of the moments from the expectations (15.261)–(15.261) of from the expectations (15.283)–(15.285) we encounter the integrals depending on ρn(s) δn + [δ(s)+ δ(s L)] /2 with δn = δ(0) σn/L or ρk(s) = δk + [δ(s)+ δ(s L)] /2 with δ =≡δ(0) ik/L: − − − k − L L n(k) ′ ′ 2 ′ H1 = ds ds ρn(k)(s)ρn(k)(s )∆ (s,s ), Z0 Z0 1 = δ2 I + δ I + ∆2(0, 0)+∆2(0,L) , (15B.1) n(k) 5 n(k) 6 2 L L I Hn(k) = ds ds′ρ (s′)∆(s,s)˙∆ 2(s,s′)= δ I + 9 , (15B.2) 2 n(k) n(k) 7 2 Z0 Z0 L L I Hn(k) = ds ds′ρ (s′)˙∆ d(s,s)∆2(s,s′)= δ I + 6 δ(0), (15B.3) 3 n(k) n(k) 5 2 Z0 Z0 L L I Hn(k) = ds ds′ρ (s′)˙∆(s,s)˙∆(s,s′)∆(s,s′)= δ I + 10 . (15B.4) 4 n(k) n(k) 8 2 Z0 Z0 When calculating Rn , we need to insert here δ = δ(0) σ /L, thus obtaining h i n − n L4 L3 L2 Hn = δ2(0)+ (3 D n) δ(0)+ (45 24D+4D2) 4n(6 2D n) , (15B.5) 1 90 45 − − 360 − − − − L3 L2 Hn = δ(0) + (15 4D 4n), (15B.6) 2 45 180 − − L4 L3 Hn = δ2(0)+ (3 D n)δ(0), (15B.7) 3 90 90 − − L3 L2 Hn = δ(0)+ (21 4D 4n) , (15B.8) 4 180 720 − − where the values for n = 0 correspond to the partition function Z = R0 . The substitution δ = δ(0) ik/L required for the calculation of P (k; L) yields h i k − L4 L3 L2 5L2 Hk = δ2(0) + [2 + ( ik)] δ(0) + ( ik) [4 + ( ik)] + , (15B.9) 1 90 45 − 90 − − 72 L3 L2 Hk = δ(0) + [11 + 4( ik)] , (15B.10) 2 45 180 − L4 L3 Hk = δ2(0) + [2 + ( ik)] δ(0), (15B.11) 3 90 90 − L3 L2 Hk = δ(0) + [17 + 4( ik)] . (15B.12) 4 180 720 − The other loop integrals are

L L L3 H = ds ds′∆(s,s)˙∆ 2(s,s′)˙∆ d(s′,s′)= δ(0)I = δ(0), (15B.13) 5 7 45 Z0 Z0 1092 15 Path Integrals in Polymer Physics

L L L4 H = ds ds′˙∆ d(s,s)∆2(s,s′)˙∆ d(s′,s′)= δ2(0)I = δ2(0), (15B.14) 6 5 90 Z0 Z0 L L L3 H = ds ds′˙∆(s,s)˙∆(s,s′)∆(s,s′)˙∆ d(s′,s′)= δ(0)I = δ(0), (15B.15) 7 8 180 Z0 Z0 L L L2 H = ds ds′˙∆(s,s)˙∆(s,s′) ∆˙(s,s′)˙∆(s′,s′)= , (15B.16) 8 −720 Z0 Z0 L L I I 7L2 H = ds ds′˙∆(s,s)∆(s,s′)˙∆ d(s,s′)˙∆(s′,s′)= 10 8 H = , (15B.17) 9 2 − L − 8 360 Z0 Z0 L L I I 7L2 H = ds ds′∆(s,s)˙∆(s,s′)˙∆ d(s,s′)˙∆(s′,s′)= 9 7 = , (15B.18) 10 4 − 2L 360 Z0 Z0 L L I 2(I I ) H = ds ds′∆(s,s)˙∆ d2(s,s′)∆(s′,s′)= δ(0)I + ( 1 )2 3 − 11 2I , 11 3 L − L − 4 Z0 Z0 L3 L2 +2 ∆2(L,L) ∆˙(L,L) ∆2(0, 0) ∆˙(0, 0) 2H = δ(0) + , (15B.19) − − 10 30 9 L L L2 H = ds ds′˙∆ 2(s,s′) ∆˙2(s,s′)= , (15B.20) 12 90 Z0 Z0 L L I I H L2 H = ds ds′∆(s,s′)˙∆(s,s′) ∆˙(s,s′)˙∆ d(s′,s)= 4 12 12 = , (15B.21) 13 2 − 2L − 2 −720 Z0 Z0 L L 2(I I ) I H = ds ds′∆2(s,s′)˙∆ d2(s,s′)= δ(0)I 3 − 12 2I + 5 , 14 3 − L − 4 L2 Z0 Z0 L3 11L2 +2 ∆2(L,L) ∆˙(L,L) ∆2(0, 0) ∆˙(0, 0) 2H = δ(0) + . (15B.22) − − 13 30 72 Appendix 15C Integrals Involving Modiﬁed Green Func- tion

To demonstrate the translational invariance of results showed in the main text we use the slightly modified Green function L s s′ (s + s′) (s2 + s′2) ∆(s,s′)= a | − | + , (15C.1) 3 − 2 − 2 2L containing an arbitrary constant a. The following combination yields the standard Feynman propagator for the infinite interval [compare (3.249)] 1 1 1 ∆ (s,s′) = ∆ (s s′) = ∆(s,s′) ∆(s,s) ∆(s′,s′)= (s s′). (15C.2) F F − − 2 − 2 −2 − Other useful relations fulfilled by the Green function (15C.1) are, assuming s s′, ≥ s(L s) (s s′)(s+s′)2 La D (s,s′) = ∆2(s,s′) ∆(s,s)∆(s′,s′) = (s s′) − + − , 1 − − L 4L2 − 3 (15C.3) (s s′)(s+s′ L) D (s,s′) = ∆(s,s) ∆(s′,s′)= − − . (15C.4) 2 − L The following integrals are needed:

L (s4 +s′4) (2s3 +s′3) J (s,s′) = dt ∆(t,t)˙∆(t,s)˙∆(t,s′)= , 1 4L2 − 3L Z0 ((a+3)s2 +as′2) sa (20a 9) + L+ − L2, (15C.5) 6 − 3 180

H. Kleinert, PATH INTEGRALS Notes and References 1093

L ( 2s4 +6s2s′2 +3s′4) J (s,s′) = dt ˙∆(t,t)∆(t,s)˙∆(t,s′)= − , 2 24L2 Z0 (3s3 3s2s′ 6ss′2 s′3) (5s2 12ss′ 4as′2) s′a (20a 3) + − − − − − L+ − L2, 12L − 24 − 6 720 (15C.6) L (s4 +6s2s′2 +s′4) (s2 +3s′2)s J (s,s′) = dt ∆(t,s)∆(t,s′)= + , 3 − 60L 6 Z0 (s2 +s′2) (5a2 10a+6) L+ − L3. (15C.7) − 6 45 These are the building blocks for other relations: (d 1) (d+1) (d 1)(s s′) f (q(s), q(s′)) = − D (s,s′) D2(s,s′) = − − h 2 i 2 1 − 4 2 − 4 2La (d 3)s (d+1)s′ (d 1)s2 (d+1)s′2 d(s s′)(s+s′)2 + − − − − + − , (15C.8) × 3 2 − L 2L2 and 1 1 K (s,s′) = J (s,s′) J (s,s) J (s′,s′), 1 1 − 2 1 − 2 1 La (s+s′) (s2 +ss′ +s′2) = (s s′) + , (15C.9) − − 6 − 4 6L K (s,s′) = J (s,s′)+J (s′,s) J (s,s) J (s′,s′), 2 2 2 − 2 − 2 1 (5s+7s′) (s+s′)2 = (s s′)2 + , (15C.10) − − 4 − 12L 4L2 1 1 (s+2s′) (s+s′)2 K (s,s′) = J (s,s′) J (s,s) J (s′,s′)= (s s′)2 , (15C.11) 3 3 − 2 3 − 2 3 − − 6 − 8L 1 1 (s s′)2(s+s′ 2L)2 K (s,s′) = ∆(0,s)∆(0,s′) ∆2(0,s) ∆2(0,s′)= − − , (15C.12) 4 − 2 − 2 − 8L2 1 1 (s2 s′2)2 K (s,s′) = ∆(L,s)∆(L,s′) ∆2(L,s) ∆2(L,s′)= − . (15C.13) 5 − 2 − 2 − 8L2 Notes and References

Random chains were ﬁrst considered by K. Pearson, Nature 77, 294 (1905) who studied the related problem of a random walk of a drunken sailor. A.A. Markoﬀ, Wahrscheinlichkeitsrechnung, Leipzig 1912; L. Rayleigh, Phil. Mag. 37, 3221 (1919); S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). The exact solution of PN (R) was found by L.R.G. Treloar, Trans. Faraday Soc. 42, 77 (1946); K. Nagai, J. Phys. Soc. Japan, 5, 201 (1950); M.C. Wang and E. Guth, J. Chem. Phys. 20, 1144 (1952); C. Hsiung, A. Hsiung, and A. Gordus, J. Chem. Phys. 34, 535 (1961). The path integral approach to polymer physics has been advanced by S.F. Edwards, Proc. Phys. Soc. Lond. 85, 613 (1965); 88, 265 (1966); 91, 513 (1967); 92, 9 (1967). See also H.P. Gilles, K.F. Freed, J. Chem. Phys. 63, 852 (1975) and the comprehensive studies by K.F. Freed, Adv. Chem. Phys. 22,1 (1972); 1094 15 Path Integrals in Polymer Physics

K. It¯oand H.P. McKean, Diﬀusion Processes and Their Simple Paths, Springer, Berlin, 1965.

An alternative model to stimulate stiffness was formulated by A. Kholodenko, Ann. Phys. 202, 186 (1990); J. Chem. Phys. 96, 700 (1991) exploiting the statistical properties of a Fermi field. Although the physical properties of a stiff polymer are slightly misrepresented by this model, the distribution functions agree quite well with those of the Kratky-Porod chain. The advantage of this model is that many properties can be calculated analytically.

The important role of the upper critical dimension Duc = 4 for polymers was first pointed out by R.J. Rubin, J. Chem. Phys. 21, 2073 (1953). The simple scaling law R2 L6/(D+2), D< 4 was first found by P.J. Flory, J. Chem. Phys.h i∝17, 303 (1949) on dimensional grounds. The critical exponent ν has been deduced from computer enumerations of all self-avoiding polymer configurations by C. Domb, Adv. Chem. Phys. 15, 229 (1969); D.S. Kenzie, Phys. Rep. 27, 35 (1976). For a general discussion of the entire subject, in particular the field-theoretic aspects, see P.G. DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, N.Y., 1979. The relation between ensembles of random chains and field theory is derived in detail in H. Kleinert, Gauge Fields in Condensed Matter, World Scientific, Singapore, 1989, Vol. I, Part I; Fluctuating Fields and Random Chains, World Scientific, Singapore, 1989 (http://www.physik.fu-berlin.de/~kleinert/b1).

The particular citations in this chapter refer to the publications [1] The path integral (15.112) for stiff polymers was proposed by O. Kratky and G. Porod, Rec. Trav. Chim. 68, 1106 (1949). The lowest moments were calculated by J.J. Hermans and R. Ullman, Physica 18, 951 (1952); N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Japan 22, 219 (1967), and up to order 6 by R. Koyama, J. Phys. Soc. Japan, 34, 1029 (1973). Still higher moments are found numerically by H. Yamakawa and M. Fujii, J. Chem. Phys. 50, 6641 (1974), and in a large-stiffness expansion in T. Norisuye, H. Murakama, and H. Fujita, Macromolecules 11, 966 (1978). The last two papers also give the end-to-end distribution for large stiffness. [2] H.E. Stanley, Phys. Rev. 179, 570 (1969). [3] H.E. Daniels, Proc. Roy. Soc. Edinburgh 63A, 29 (1952); W. Gobush, H. Yamakawa, W.H. Stockmayer, and W.S. Magee, J. Chem. Phys. 57, 2839 (1972). [4] B. Hamprecht and H. Kleinert, J. Phys. A: Math. Gen. 37, 8561 (2004) (ibid.http/345). Mathematica program can be downloaded from ibid.http/b5/prm15. [5] The Mathematica notebook can be obtained from ibid.http/b5/pgm15. The program needs less than 2 minutes to calculate R32 . [6] J. Wilhelm and E. Frey, Phys. Rev. Lett. 77, 2581 (1996). See also: A. Dhar and D. Chaudhuri, Phys. Rev. Lett 89, 065502 (2002); S. Stepanow and M. Schuetz, Europhys. Lett. 60, 546 (2002); J. Samuel and S. Sinha, Phys. Rev. E 66 050801(R) (2002).

H. Kleinert, PATH INTEGRALS Notes and References 1095

[7] H. Kleinert and A. Chervyakov, Perturbation Theory for Path Integrals of Stiff Polymers, Berlin Preprint 2005 (ibid.http/358). [8] For a detailed theory of such fields with applications to superconductors and superfluids see H. Kleinert, Collective Quantum Fields, Fortschr. Phys. 26, 565 (1978) (ibid.http/55). [9] S.F. Edwards and P.W. Anderson, J. Phys. F: Metal Phys. 965 (1975). Applications of replica field theory to the large-L behavior of R2 and other critical exponents were given by h i P.G. DeGennes, Phys. Lett. A 38, 339 (1972); J. des Cloizeaux, Phys. Rev. 10, 1665 (1974); J. Phys. (Paris) Lett. 39 L151 (1978). [10] See D.J. Amit, Renormalization Group and Critical Phenomena, World Scientific, Singa- pore, 1984; G. Parisi, Statistical Field Theory, Addison-Wesley, Reading, Mass., 1988. [11] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific, Singapore, 2000 (ibid.http/b8). [12] For the calculation of such quantities within the φ4-theory in 4 ǫ dimensions up to order ǫ5, see − H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin, Phys. Lett. B 272, 39 (1991) (hep-th/9503230). [13] For a comprehensive list of the many references on the resummation of divergent ǫ- expansions obtained in the field-theoretic approach to critical phenomena, see the textbook [11]. [14] A.J. McKane, Phys. Lett. A 76, 22 (1980).